TREATISE 


SURVEYING; 

" 


IN    WHICH 


THE  THEORY  AND  PEACTICE  AEE  FULLY 
EXPLAINED. 


PRECEDED    BY 


A  SHORT  TREATISE  ON  LOGARITHMS: 


AND    ALSO    BY 


A  COMPENDIOUS  SYSTEM  OF  PLANE  TRIGONOMETRY. 

%tyt  fojjole  f  Ilttsiratefr  bg  $hromotts 


SAMUEL  ALSOP, 

\ 

AUTHOR    OP    A    TREATISE    ON    ALGEBRA,    ETC. 


THIKD  EDITION.    <•  ,  • 


PHILADELPHIA: 
E.  C.  &  J.  BIDDLE,  No.  508  MINOR  ST. 

(Between  Market  and  CTiestnut,  and  Fifth  and  Sixth  Sts.) 

1865. 


PREFACE. 


THE  favor  shown  to  this  treatise  by  the  author's  colaborers  in 
the  educational  field  having  called  for  another  edition  of  it,  he 
has  carefully  revised  the  work,  and  made  such  amendments  as 
to  him  seemed  desirable.  These  are  not  numerous,  but  he 
believes  have  somewhat  improved  the  work. 

His  aim  has  been  to  present  the  subject,  in  its  practical  as 
well  as  its  theoretical  relations,  in  a  manner  adapted  to  the  capa- 
city of  every  student,  by  presenting  the  theory  plainly  and  com- 
prehensively, and  giving  definite  and  precise  directions  for  prac- 
tice ;  and  to  embrace  in  the  work  every  thing  which  an  extensive 
business  in  land-surveying  would  be  likely  to  require.  How 
nearly  his  object  has  been  attained,  others  must  determine :  he 
trusts,  however,  that  the  treatise  will  be  found  to  possess  merit 
sufficient  to  commend  it  to  the  favorable  notice  of  his  fellow- 
teachers.  The  following  brief  synopsis  of  its  contents  presents 
the  plan  and  scope  of  the  work. 

Chapter  I.  consists  of  a  short  explanation  of  the  nature  and 

use  of  Logarithms. 

5 


f\ 


PREFACE. 

Chapter  II.  contains  the  geometrical  definitions  and  con- 
structions needed  in  the  subsequent  part  of  the  work. 

In  Chapter  III.  is  presented  a  treatise  on  Plane  Trigono- 
metry, including  a  great  variety  of  examples  illustrative  of  the 
solution  of  triangles.  In  this  chapter  will  also  be  found  a  full 
description  of  the  Theodolite  and  Surveyor's  Transit,  and  direc- 
tions for  their  use. 

In  Chapter  IY.  the  principles  of  surveying  by  the  Chain  are 
explained.  This  method  is  little  employed  by  practical  sur- 
veyors in  this  country.  Since,  however,  the  measurements 
require  no  other  instrument  than  a  tape-line,  or  a  cord,  or  some 
other  means  of  determining  distances,  it  is  of  importance  to  the 
farmer,  who  frequently  desires  to  know  the  contents  of  par- 
ticular fields,  or  of  portions  of  enclosures.  The  second  and 
third  sections  of  this  chapter  contain  a  pretty  full  treatise  on 
Field  Geometry,  or  the  method  of  performing  on  the  ground, 
with  the  chain  or  measuring  line  only,  those  operations  which 
are  needed  in  fixing  the  positions  of  points  or  in  locating  lines. 
In  Great  Britain,  Chain  Surveying  is  almost  exclusively  em- 
ployed. 

Chapter  Y.  is  devoted  to  Compass  Surveying.  Under  this 
head  are  included  all  those  methods  which  require  the  use  of  an 
instrument  for  determining  the  beariDgs  of  lines,  whether  that 
instrument  be  a  Compass,  a  Transit,  or  a  Theodolite.  This 
chapter  contains  a  full  account  of  the  methods  to  be  employed 
in  locating  lines  by  means  of  such  instruments. 

The  numerous  difficulties  with  which  the  surveyor  will  be 
likely  to  meet  from  obstructions  on  the  ground  are  stated,  and 
the  modes  of  overcoming  them  explained. 

This  chapter,  with  that  on  Plane  Trigonometry,  constitutes, 
in  fact,  a  full  treatise  on  Surveying  as  practised  in  this  country. 
In  selecting  the  methods  to  be  employed  in  overcoming  the 
difficulties  both  in  Compass  and  in  Chain  Surveying,  care  has 
been  taken  to  adopt  such  only  as  may  be  conveniently  employed 
in  the  field. 

Chapter  VI.  contains   the   general   principles   of  Triangular 


PREFACE.  7 

Surveying.  This  is  the  method  employed  in  extensive  geodetic 
operations. 

The  details  of  this  method  are  so  complex  that  a  volume — 
not  a  chapter — would  be  required  for  their  development.  All 
that  has  been  attempted  is  to  give  some  of  the  more  simple 
principles. 

Chapter  VII.  treats  of  Laying  out  and  Dividing  Land.  It  is 
believed  that  many  of  the  demonstrations  in  this  chapter  will  be 
found  to  be  much  more  simple  than  those  usually  given,  almost 
all  of  them  having  been  reduced  to  the  development  of  a  single 
principle.  On  a  subject  of  this  kind,  which  has  so  long  occupied 
the  attention  of  mathematicians,  any  thing  new  could  hardly 
be  expected.  It  has  been  the  aim  of  the  author  to  select  the 
best  methods,  not  to  introduce  any  thing  merely  because  it 
was  new. 

Chapter  IX.  contains  a  treatise  on  Practical  Astronomy, 
embracing  all  that  is  needed  for  the  surveyor's  purposes  or  is 
practicable  with  his  instruments.  Various  methods  of  running 
meridian  lines,  and  of  determining  the  latitude  and  the  time  of 
day,  are  fully  explained. 

The  concluding  chapter  (X.)  is  devoted  to  the  subject  of 
the  Variation  of  the  Compass.  In  it  will  be  found  information 
of  great  value  to  the  practical  surveyor.  The  tables  of  varia- 
tion are  in  all  cases  drawn  from  the  most  recent  and  authentic 
sources. 

In  the  preparation  of  this  treatise  the  author  has  consulted 
various  well-known  English  and  American  mathematical  works. 
To  Professor  GILLESPIE'S  excellent "  Treatise  on  Land  Surveying" 
(D.  Appleton  &  Co.,  New  York,)  especially,  the  author  is  indebted 
for  very  valuable  hints,  particularly  in  the  directions  for  prac- 
tice, the  descriptions  of  the  instruments,  and  various  new 
methods  of  presenting  important  points.  Some  of  these  are 
referred  to  in  their  places.  The  typographical  peculiarities  of 
this  volume,  in  the  headings  of  articles,  &c.,  were  also  suggested 
to  the  publishers  by  those  of  the  work  of  Dr.  Gillespie. 

In  each  department  of  the  subject  treated  of  in  this  volume 


8  PREFACE. 

the  aim  of  the  author  has  been  to  explain  clearly  the  principles 
involved,  and,  as  a  general  rule,  to  give  only  those  methods  for 
practice  which  he  deems  the  best.  By  pursuing  this  course  he 
has  kept  the  volume  within  moderate  limits,  and  has  presented, 
the  subject  in  such  a  form  as  will,  he  trusts,  meet  the  wants  of 
teachers  generally,  as  well  as  of  very  many  practical  surveyors. 
The  tables  appended  to  this  treatise  have  been  prepared  with 
much  care.  That  of  Latitudes  and  Departures  will  be  found  to 
be  more  concise  than  those  usually  given,  and,  being  extended 
to  four  decimal  places,  will  enable  the  calculator  to  give  greatei 
accuracy  to  his  work.  The  table  of  Logarithms  of  Numbers 
has  been  carefully  compared  with  those  of  Babbage,  Hutton, 
and  other  standard  authors.  That  of  Sines  and  Tangents  waa 
taken  from  Hutton,  and  compared  with  other  seven-decimal 
tables.  Besides  these,  there  is  a  table  of  Natural  Sines  and 
Cosines  to  every  minute,  and  one  of  Chords  to  every  five 
minutes,  of  the  quadrant. 


CONTENTS. 


CHAPTER  I. 

ON  THE  NATURE  AND  USE  OF  LOGARITHMS. 

SECTION  1.  On  the  Nature  of  Logarithms.  PAQA 

Definition  and  Illustration 17 

Mode  of  calculating  Logarithms 19 

Bases  of  Logarithms 19 

Indices  of  Logarithms 20 

Mantissse  of  Logarithms 20 

Description  of  the  Table  of  Logarithms 20 

To  find  the  Logarithm  of  a  Number  from  the  Table 21 

To  find  the  Natural  Number  corresponding  to  a  given  Logarithm 23 

SECTION  2.   On  the  Use  of  Logarithms. 

Multiplication  by  Logarithms 25 

Division,  by  Logarithms 26 

Involution  by  Logarithms 27 

Evolution  by  Logarithms 27 

On  the  Use  of  Arithmetical  Complements  of  Logarithms 28 

CHAPTER  II. 

PRACTICAL   GEOMETRY 

SECTION  1.  Definitions 31 

SB«TION  2.   Geometrical  Properties  and  Problems ; 36 

A.  Geometrical  Properties 36 

B.  Geometrical  Problems 39 

To  bisect  a  given  Straight  Line 39 

To  draw  a  Perpendicular  to  a  Straight  Line  from  a  Point  in  it 40 

To  let  fall  a  Perpendicular  to  a  Line  from  a  Point  without  it 40 

At  a  given  Point,  to  make  an  Angle  equal  to  a  given  Angle 41 

To  bisect  a  given  Rectilineal  Angle 42 

To  draw  a  Straight  Line  touching  a  Circle 42 

Through  a  given  Point  to  draw  a  Parallel  to  a  given  Straight  Line 42 

To  inscribe  a  Circle  in  a  given  Triangle 43 

To  describe  a  Circle  about  a  given  Triangle 43 

To  find  a  Third  Proportional  to  two  Straight  Lines 43 

To  find  a  Fourth  Proportional  to  three  Straight  Lines 43 

To  find  a  Mean  Proportional  between  two  Straight  Lines 44 

To  divide  a  Line  into  two  Parts  having  a  given  Ratio 44 


10  CONTENTS. 

» 

CHAPTER  III. 

PLANE    TKIGONOMETRY. 

SECTION  1.  Definitions.  PAGE 

Measure  of  Angles 45 

Trigonometrical  Functions 46 

Properties  of  Sines,  Tangents,  &c - 47 

Geometrical  Properties  employed  in  Plane  Trigonometry 48 

SECTION  2.  Drafting  or  Platting. 

Mode  of  drawing  Straight  Lines 49 

Mode  of  drawing  Parallels 49 

Mode  of  drawing  Perpendiculars 51 

Mode  of  drawing  Circles  and  Arcs 51 

Mode  of  laying  off  Angles  with  a  Protractor 52 

By  a  Scale  of  Chords 52 

By  a  Table  of  Chords 53 

Distances 53 

Drawing  to  a  Scale 53 

Scales 55 

Diagonal  Scale 55 

Proportional  Scale 57 

Vernier  Scale 57 

SECTION  3.   Tables  of  Trigonometrical  Functions. 

Description  of  the  Table  of  Natural  Sines  and  Cosines 58 

Description  of  the  Table  of  Logarithmic  Sines  and  Tangents 59 

Use  of  Table 60 

Table  of  Chords 63 

SECTION  4.   On  the  Numerical  Solution  of  Triangles. 

Definition 64 

The  Numerical  Solution  of  Right-Angled  Triangles 64 

By  the  Use  of  the  Table  of  Sines  and  Tangents 64 

By  the  Application  of  (47.1.) 66 

The  Numerical  Solution  of  Oblique- Angled  Triangles. 

The  Angles  and  one  Side,  or  two   Sides  and  an  Angle  opposite  one 

of  them,  being  given,  to  find  the  rest 67 

Two  Sides  and  the  included  Angle  being  given,  to  find  the  rest. 

Rulel 70 

Rule  2 71 

The  three  Sides  being  given,  to  find  the  Angles. 

Rulel 73 

Rule  2 74 

SECTION  5.  Instruments,  and  Field  Operations. 

The  Chain 76 

The  Pins 78 

Chaining , 78 

Recording  the  Outs 79 

Horizontal  Measurement 80 

Tape  Lines 82 

Angles 82 

The  Transit  and  Theodolite. 

General  Description 83 

The  Telescope 87 


CONTENTS.  .  11 

PAGE 

The  Object  Glass 88 

The  Eye  Piece 88 

The  Spider  Lines 89 

The  Supports 91 

The  Vertical  Limb 91 

The  Levels 92 

The  Levelling  Plates 92 

The  Clamp  and  Tangent  Screws...., 93 

The  Watch  Telescope 

Verniers , 93 

The  Reading  of  the  Vernier 95 

To  Read  any  Vernier , 96 

Retrograde  Verniers 96 

Reading  backwards 98 

Double  Verniers 98 

Adjustments 101 

First  Adjustment:  The  Level  should  be  parallel  to  the  Horizontal  Plates  102 
Second  Adjustment :  The  Axis  of  the  Horizontal  Plates  should  be  pa- 
rallel   102 

Third  Adjustment:  The  Line  of  Collimation  must  be  perpendicular 

to  the  Horizontal  Axis 102 

The  Line  of  Collimation  in  the  Theodolite  should  be  parallel  to  the 

Axis  of  the  Cylinders  on  which  the  Telescope  rests  in  its  Ys 104 

Fourth  Adjustment:   The  Horizontal  Axis  must  be  parallel  to  the 

Horizontal  Plates 104 

Adjustments  of  the  Vertical  Limb 105 

First  Adjustment :  The  Level  must  be  parallel  to  the  Line  of  Colli- 
mation    105 

Second  Adjustment :    The  Zeros  of  the  Vernier  and  Vertical  Limb 

should  coincide  when  the  Telescope  is  horizontal 106 

Measuring  Angles 107 

Repetition  of  Angles 108 

Verification  of  Angles 109 

Reduction  to  the  Centre 109 

Angles  of  Elevation 110 

SECTION  6.  Miscellaneous  Problems  to  Illustrate  the  Rules  of  Plane  Trigono- 
metry   110 

CHAPTER   IV. 

CHAIN     SURVEYING. 

SECTION  1.  Definitions. 

Definition 118 

Advantages 118 

Area  Horizontal , 119 

SECTION  2.  Field  Operations. 

Ranging  out  Lines •  119 

To  Interpolate  Points  in  a  Line 120 

On  Level  Ground 120 

Over  a  Hill 120 

By  a  Random  Line 121 

Across  a  Valley 122 

To  determine  the  Point  of   Intersection  of  two  visual  Lines 123 

To  run,  a  Line  towards  an  invisible  Intersection 123 

Perpendiculars. 

To  draw  a  Perpendicular  to  a  given  Line  from  a  Point  in  it. 

When  the  Point  is  accessible 123 


12  CONTENTS. 

PAGH 

When  the  Point  is  inaccessible 125 

To  let  fall  a  Perpendicular  to  a  Line  from  a  point  without  it. 

When  the  Point  and  Line  are  both  accessible 125 

When  the  Point  is  remote  or  inaccessible 126 

When  the  Line  is  inaccessible 126 

The  Surveyor's  Cross 127 

To  verify  the  Cross 128 

The  Optical  Square 128 

To  test  the  Accuracy  of  the  Square 129 

Parallels 

Through  a  given  Point  to  draw  a  Parallel  to  an  accessible  Line 130 

To  draw  a  Parallel  to  an  inaccessible  Line 130 

To  draw  a  Parallel  to  a  Line  through  an  inaccessible  Point 130 

SECTION  3.   Obstacles  in  Running  and  Measuring  Lines. 

To  prolong  a  Line  beyond  an  Obstacle 131 

To  measure  a  line  when  both  ends  are  accessible 132 

When  one  End  is  inaccessible 133 

When  the  inaccessible  End  is  the  intersection  of  two  Lines 133 

When  both  Ends  are  inaccessible 134 

SECTION  4.  Keeping  Field  Notes , 135 

Field  Book 135 

Test  Lines 139 

General  Directions 139 

Platting  the  Survey 140 

SECTION  5.  Surveying  Fields  of  Particular  Form. 

Rectangles 141 

Parallelograms 141 

Triangles. 

First  Method 142 

Second  Method 142 

Trapezoids 144 

Trapeziums. 

First  Method 145 

Second  Method 145 

Fields  of  more  than  four  Sides. 

First  Method 147 

Second  Method 150 

Offsets 151 

SECTION  6.  Tie  Lines. 

Inaccessible  Areas 159 

Defects  of  the  Method 159 


CHAPTER  V. 

COMPASS    SURVEYING. 

SECTION  1.  Definitions  and  Instruments. 

The  Meridian 160 

The  Points  of  the  Compass 161 

Bearing 161 

Reverse  Bearing 162 

The  Magnetic  Needle 162 

The  Magnetic  Meridian 163 


CONTENTS.  13 

PAGS 

The  Magnetic  Bearing 163 

The  Compass... 164 

The  Sights * 166 

The  Verniers 166 

The  Pivot 168 

The  Divided  Circle 168 

Adjustments -. 169 

Defects  of  the  Compass 169 

SECTION  2.  Field  Operations. 

Bearings 170 

Use  of  the  Vernier 171 

The  Reverse  Bearing 171 

Local  Attraction 171 

To  correct  for  Back  Sights 172 

By  the  Vernier 172 

To  survey  a  Farm — General  Directions., 172 

Random  Line 173 

To  determine  the  Bearing  by  a  Station  near  the  Middle  of  the  Line 174 

Proof  Bearings 174 

Angles  of  Deflection 175 

SECTION  3.    Obstacles  in  Compass  Surveying. 

To  run  a  Line  making  a  given  Angle  with  a  given  Line   at  a  given 

Point  within  it 176 

To  run  a  Line  making  a  given  Angle  with  a  given  inaccessible  Line  at  a 

given  Point  in  that  Line 177 

From  a  given  Point  out  of  a  Line,  to  run  a  Line  making  a  given  Angle 
with  that  Line. 

If  the  Line  be  accessible 177 

If  the  Line  be  inaccessible 178 

If  the  Point  be  inaccessible, 178 

If  the  Point  and  the  Line  be  both  inaccessible , 179 

To  run  a  Line  parallel  to  a  given  Line  through  a  given  Point. 

If  the  Line  and  the  Point  be  accessible 179 

If  the  Point  be  inaccessible 179 

If  the  Line  be  inaccessible 179 

If  the  Line  and  the  Point  both  be  inaccessible 180 

Prolongation  and  Interpolation  of  Lines 180 

To  Prolong  a  Line  beyond  an  Obstruction 181 

To  Interpolate  Points  in  a  Line 182 

By  a  Random  Line 182 

Measurement  of  Distances. 

To  determine  the  Distance  between  two  Points  visible  from  each 

other.... 183 

To  determine  the  Distance  on  a  Line  to  the  inaccessible  but  visible 

end 185 

To  determine  the  Distance  when  the  end  is  invisible 186 

To  determine  the  Distance  to  the  Intersection  of  two  Lines..-. 186 

To  determine  the  Distance  between  two  inaccesible  Points 187 

Examples  illustrative  of  the  preceding  Rules 188 

SECTION  4.  Field  Notes 190 

SECTION  5.  Latitudes  and  Departures. 

Definitions 192 

The  Bearing,  Distance,  Latitude,  and  Departure, — any  two  being  given, 

to  determine  the  others 193 

To  determine  the  Latitude  and  Departure  by  the  Traverse  Table 194 

When  the  Bearing  is  given  by  Minutes 196 


14  CONTENTS. 

PAGE 

By  the  Table  of  Natural  Sines  and  Cosines 197 

Test  of  the  Accuracy  of  the  Survey 199 

Correction  of  Latitudes  and  Departures 200 

SECTION  6.  Platting  the  Survey. 

With  the  Protractor 202 

By  a  Scale  of  Chords 203 

By  a  Table  of  Natural  Sines 204 

By  a  Table  of  Chords 205 

By  Latitudes  and  Departures 205 

SECTION  7.  Problems  in  Compass  Surveying. 

Given  the  Bearing  of  one  Side,  and  the  Deflection  of  the  next,  to  deter- 
mine its  Bearing 208 

To  determine  the  Deflection  between  two  Courses 209 

To  determine  the  Angle  between  two  Lines 210 

To  change  the  Bearings  of  the  Sides  of  a  Survey 211 

SECTION  8.  Supplying  Omissions. 

The  Bearings  and  Distances  of  all  the  Sides  except  one  being  given,  to 

determine  these 213 

All  the  Bearings  and  Distances  except  the  Bearing  of  one  Side  and 

the  Distance  of  another  being  given,  to  find  these 217 

All  the  Bearings  and  Distances  except  two  Distances  being  given,  to  find 

these 219 

All  the  Bearings  and  Distances  except  two  Bearings  being  given,  to  find 

these 220 

SECTION  9.    Content  of  Land. 

Given  two  Sides  and  the  included  Angle  of  a  Triangle  or  Parallelogram, 

to  find  the  Area 224 

The  Angles  and  one  Side  of  a  Triangle  being  given,  to  find  the  Area 225 

To  determine  the  Area  of  a  Trapezium,  three  Sides  and  the  two  included 

Angles   being  given 226 

The  Bearings  and  Distances  of  the  Sides  of  a  Tract  of  Land  being 

given,  to  find  its  Area 229 

Offsets 235 

Inaccessible  Areas 238 

Compass  Surveying  by  Triangulation 243 

CHAPTER  VI. 

TRIANGULAR    SURVEYING. 

Base 247 

Reduction  to  the  Level  of  the  Sea 248 

Signals 248 

Triangulation 248 

Base  of  Verification 250 


CHAPTER  VIL 

LAYING   OUT   AND   DIVIDING   LAND. 

SECTION  1.  Laying  out  land. 

To  lay  out  a  given  Quantity  of  Land  in  the  form  of  a  Square 251 

To  lay  out  a  given  Quantity  of  Land  in  the  form  of  a  Rectangle,  one  Side 

being  given 251 

The  Adjacent  Sides  having  a  given  Ratio 252 


CONTENTS.  15 

PAGE 

One  Side  to  exceed  another  by  a  given  Difference 252 

To  lay  out  a  given  quantity  of  Land  in  the  form  of  a  Triangle  or  Paral- 
lelogram, the  Base  being  given 253 

One  Side  and  the  Adjacent  Angle  being  given 253 

Lemma 254 

The  Direction  of  two  Adjacent  Sides  being  given,  to  lay  out  a  given 
quantity  of  land. 

By  a  Line  running  a  given  Course , 255 

By  a  Line  running  through  a  given  Point 256 

Three  Adjacent  Sides  of  a  Tract  being  given  in  Position,  to  lay  out  a 

given  quantity  of  land 259 

By  a  Line  parallel  to  the  second  Side 259 

By  a  Line  running  a  given  Course 262 

By  a  Line  througha  given  Point 267 

By  the  shortest  Line 269 

To  cut  off  a  Plat  containing  a  given  Area  from  a  Tract  of  any  number  of 
Sides. 

By  a  Division  line  drawn  from  one  of  the  Angles 269 

By  a  Line  running  a  given  Course 273 

To  straighten  Boundary  lines 275 

To  run  a  new  Line  between  Tracts  of  different  Values. 

By  a  Line  running  a  given  Course 280 

By  a  Line  through  a  given  Point  in  the  old  Line 281 

By  a  Line  through  a  given  Point  in  one  of  the  Adjacent  Sides 283 

SECTION  2.  Division  of  Land. 

To  divide  a  Triangle  into  two  Parts  having  a  given  Ratio. 

By  a  Line  through  one  of  the  Corners 284 

By  a  Line  through  a  Point  in  one  of  the  Sides 284 

By  a  Line  Parallel  to  one  of  the  Sides 286 

By  a  Line  running  a  given  Course 286 

By  a  Line  through  a  given  Point 288 

To  divide  a  Trapezoid  into  two  parts  having  a  given  Ratio. 

By  a  Line  cutting  the  Parallel  Sides 290 

By  a  Line  Parallel  to  the  Parallel  Sides 292 

To  divide  a  Trapezium  into  two  parts  having  a  given  Ratio. 

By  a  Line  through  a  given  Point  on  one  Side 294 

By  a  Line  through  any  Point 296 

By  a  Line  Parallel  to  one  Side 298 

By  a  Line  running  a  given  Course , 301 

CHAPTER  VIII. 

MISCELLANEOUS    EXAMPLES. 

Miscellaneous  Examples 303 

CHAPTER  IX. 

MERIDIANS,  LATITUDE,  AND    TIME. 

SECTION  1.  Meridians. 

Definition 307 

To  run  a  Meridian  Line. 

By  equal  Altitudes  of  the  Sun 308 

By  a  Meridian  Altitude  of  Polaris 309 

To  determine  the  Time  Polaris  is  on  the  Meridian 310 

To  run  a  Meridian  by  a  Meridian  Passage  observed  with  a  Transit  or 
Theodolite ..  314 


16  CONTENTS. 

PAGE 

By  an  Observation  of  Polaris  at  its  greatest  Elongation 314 

By  Equal  Altitudes  of  a  Star 318 

SECTION  2.  Latitude. 

To  determine  the  Latitude  by  a  Meridian  Altitude  of  Polaris 319 

By  a  Meridian  Altitude  of  the  Sun 319 

By  an  Observation  on  a  Star  in  the  Prime  Vertical 320 

SECTION  3.  To  find  the  Time  of  Day. 

By  a  Meridian  Line 322 

By  an  observed  Meridian  Passage  of  a  Star 322 

By  an  Altitude  of  the  Sun  or  a  Star  not  in  the  Meridian 323 

CHAPTER  X. 

VARIATION    OF    THE    COMPASS. 

Secular  Change 325 

Table  of  Variations 326 

Line  of  no  Variation 326 

To  determine  the  Change  in  Variation  by  old  Lines 327 

Diurnal  Changes 329 

Irregular  Changes \ 329 

APPENDIX. 

Demonstration  of  the  Rule  for  finding  the  Area  of  a  Triangle  when  three 

Sides  are  given 332 


A 

TREATISE  ON  SURVEYING, 


CHAPTER  I. 

ON    THE    NATURE   AND    USE   OF    LOGARITHMS. 


SECTION  I. 
ON  THE  NATURE  OF  LOGARITHMS. 

1.  Definition.    LOGARITHMS  are  a  series  of  numbers,  by 
the  aid  of  which  the  operations  of  multiplication,  division, 
the  raising  of  powers,  and  the  extraction  of  roots,  may, 
respectively,  be  performed  by  addition,  subtraction,  multi- 
plication, and  division. 

Such  a  series  may  be  thus  constructed.  Above  a  geometric 
series,  the  first  term  of  which  is  1,  place  a  corresponding 
arithmetic  series,  the  first  term  of  which  is  0 ;  thus : — 

Arithmetical  series,     0123456       7        8 
Geometrical  series,     1     2    4    8     16     32     64     128     256 

To  determine  the  product  of  any  two  terms  of  the  geometric 
series,  it  is  evidently  only  necessary  to  add  the  correspond- 
ing terms  of  the  arithmetic  series,  and  to  notice  the  term  of 
the  geometric  series  agreeing  to  their  sum ;  which  term  is  the 
product  required.  Thus,  to  find  the  product  of  4  and  32,  we 
add  the  corresponding  terms,  2  and  5,  in  the  arithmetic  series. 
Their  sum,  7,  corresponds  to  128,  the  product  required. 

2.  In  a  table  of  logarithms,  the  terms  of  the  geometrical 
series  are  called  the  numbers;  the  ratio  in  this  series  is  de- 
nominated the  base  of  the  table ;  and  the  terms  of  the  arith- 
metical series  are  called  the  logarithms  of  the  corresponding 

2  17 


18 


THE  NATURE  AND   USE   OF  LOGARITHMS.         [CHAP.  I. 


terms  of  the  geometric  series.  The  numbers,  it  will  be 
observed,  are  the  powers  of  the  base,  and  the  logarithms  are 
the  indices  of  those  powers. 

Further  to  illustrate  the  use  of  logarithms,  we  give  the 
following  table : — 


Num. 

Log. 

Num. 

Log. 

Num. 

Log. 

2 
4 
8 
16 
32 

1 

2 
3 
4 
5 

64 
128 
256 
512 
1024 

6 
7 
8 
9 
10 

2048 
4096 
8192 
16384 

32768 

11 

12 
13 
14 
15 

1.  Eequired  the  quotient  of  32768  divided  by  2048.     The 
indices  or  logarithms  of  these  numbers  are,  respectively,  15 
and  11.     The  difference  of  these  logarithms  is  4,  which  is 
the  logarithm  of  16,  the   quotient  required.      Hence  the 
difference  of  the  logarithms  of  two  numbers  is  the  logarithm 
of  their  quotient. 

2.  Required  the  third  power  of  32.     The  logarithm  of  32 
is  5.     Multiply  this  by  3,  the  index  of  the  power  to  which 
32  is  to  be  raised,  and  the  product,  15,  is  the  index  of  32768, 
the  required  power.    Hence,  to  involve  a  number  to  a  given 
power,  we  multiply  its  logarithm  by  the  index  of  the  power 
to  which  it  is  to  be  raised. 

3.  Required  the  fourth  root  of  4096.     The  index  of  this 
is  12.     Divide  this  index  by  4,  the  degree  of  the  root  to  be 
extracted,  and  the  quotient  will  be  3,  which  is  the  logarithm 
of  8,  the  root  required.     Hence,  to  extract  the  root  of  a 
number,  we  divide  its  logarithm  by  the  number  expressing 
the  degree  of  the  root  to  be  extra6ted,  and  the  quotient  is 
the  logarithm  of  the  root  required. 

3.  The  table  in  Art.  2  contains  only  the  integral  powers 
of  2,  that  being  sufficient  for  the  purpose  of  illustra- 
tion; but  a  complete  table  contains  all  the  numbers 
of  the  natural  series,  within  the  limits  of  the  table, 
together  with  the  indices,  or  logarithms.  The  logarithms 
in  such  a  table  will,  in  most  instances,  be  fractions. 
Thus,  the  logarithms  corresponding  to  any  of  the  num- 
bers between  4  and  8  would  be  2  and  some  fraction ; 


SEC.  I.]  THE  NATURE   OF  LOGARITHMS.  19 

of  any  number  between  8  and  16,  the  logarithm  would  be 
3  and  a  fraction ;  and  so  on. 

4.  Calculation  of  Logarithms.     Since  all  numbers  are 
considered  as  the  power  of  some  one  base,  we  will  have, 
if  a  be  the  base,  and  n  the  number,  ax  =  n.     The  deter- 
mination of  the  logarithm  will  then  consist  in  solving  the 
above  equation  so  as  to  find  x.     This,  in  general,  can  only 
be  done  by  approximation.     The  details  to  which  it  would 
lead  are  entirely  foreign  to  the  present  work.     Those  who 
desire  to  become  acquainted  with  the  subject  may  consult 
the  author's  "  Treatise  on  Algebra." 

5.  Bases.     Theoretically,  it  is  of  no  importance  what 
number  is  assumed  as  the  base  of  the  system ;  but  prac- 
tical convenience  suggests  that  10,  the  base  of  our  system 
of  notation,  should  also  be  the  base  of  the  system  of  loga- 
rithms.    By  the  use  of  this  base,  it  becomes  unnecessary 
to  insert  in  the  table  of  logarithms  their  integral  portions. 
For,  as  will  be  seen  hereafter,  the  figures  in  the  decimal  por- 
tion of  the  logarithm  depend  on  the  figures  in  the  number, 
while  the  integral  portion  of  the  logarithm  depends  solely 
on  the  position  of  the  decimal  point  in  the  number. 

6.  Assuming,  then,  10  for  a  base,  we  have  the  following 
series : — 

lumbers,        1,  10,  100,  1000,  10000,  100000,  1000000; 
Logarithms,     01234  5  6. 

The  logarithm  of  any  number  between  1  and  10  will  be 
wholly  decimal;  between  10  and  100,  it  will  be  1  and  a 
decimal ;  and  so  on. 
If  the  powers  of  10  be  continued  downwards,  we  have 

the  powers          1      .1      .01       .001       .0001      .00001, 
and  indices          0    —1     —2       —3         —4  —5. 

The  logarithm  of  any  number  between  .1  and  1  is  there- 
fore — 1  +  a  decimal,  of  a  number  between  .01  and  .1  it  is 
— 2  -f  a  decimal,  &c. 


20  THE  NATURE  AND  USE   OF  LOGARITHMS.         [CHAP.  I. 

7.  Indices  of  Logarithms.     The  integral  portion  of 
every  logarithm  is  called  the  index,  the  decimal  portion 
being   sometimes   called  the   mantissa.     From  the   above 
series,  it  is  manifest  that,  if  the  number  is  greater  than  1, 
the  index  is  positive,  and  one  less  than  the  number  of  in- 
tegral figures.     Thus,   246.75   coming  between   100   and 
1000,  its  logarithm  will  be  2  and  a  decimal.     If  the  num- 
ber is  less  than  1,  the  index  will  be  negative.     For  ex- 
ample, the  logarithm  of  .0024675,  which  comes  between 
.001  and  .01,  will  be  — 3  +  a  decimal. 

8.  Mantissse.     The  mantissas  of  logarithms  to  the  base 
10  depend  solely  on  the  figures  of  the  number,  without 
any  regard  to  the  position  of  the  decimal  point. 

Let  the  logarithm  of  31.416  be  1.497151:  then,  since 
314.16  is  10  times  31.416,  its  logarithm  will  be  1.497151  + 
1  =  2.497151.  Similarly,  the  logarithm  of  31416,  which  is 
1000  times  31.416,  will  be  1.497151  +  3  =  4.497151. 

Again,  .031416  =  31.416  -f-  1000 :  its  logarithm  is  there- 
fore 1.497151  -  3  =  —2.497151,  in  which  the  sign  —  is 
understood  to  belong  solely  to  the  index  2,  and  not  to  the 
mantissa.  Since,  then,  the  index  can  be  supplied  by  atten- 
tion to  the  position  of  the  decimal  point,  the  mantissse 
alone  are  inserted  in  the  body  of  a  table  of  logarithms. 

The  annexed  table  will  illustrate  the  above  more  fully : — 

Number.  Logarithm. 

64790  4.811508 

6479  3.811508 

647.9  2.811508 

64.79  1.811508 

6.479  0.811508 

.6479  —1.811508 

.06479  —2.811508 

.006479  —3.811508. 

9.  Table  of  Logarithms.  A  table  of  logarithms  consists 
of  the  series  of  natural  numbers,  with  their  logarithms,  or, 
rather,  the  mantissas  of  their  logarithms,  so  arranged  that 


SEC,  L] 


THE  NATURE  OF   LOGARITHMS. 


21 


one  can  be  readily  determined  from  the  other.  In  the 
table  of  logarithms  appended  to  this  treatise,  the  mantissse 
of  the  logarithms  of  all  numbers,  from  1  to  9999  inclusive, 
are  given.  On  the  first  page  are  found  the  numbers  from 
1  to  99,  with  their  logarithms  in  full.  The  remaining  pages 
contain  only  the  mantissse  of  the  logarithms.  The  first 
column,  headed  $T,  contains  the  numbers,  from  100  to  999 ; 
and  the  second,  headed  0,  the  mantissse  of  their  logarithms. 
Thus,  the  logarithm  of  the  number  897  is  2.952792;  the 
index  being  2,  because  there  are  three  integral  figures  in 
the  number. 

The  remaining  columns  contain  the  last  four  figures  of 
the  mantissse  of  the  logarithms  of  numbers  of  four  figures, 
the  first  three  of  which  are  found  in  the  first  column,  and 
the  fourth,  at  the  head.  Thus,  if  the  number  were  8976, 
the  last  four  figures  3083  of  the  mantissa  of  its  loga- 
rithm would  be  found  in  the  column  headed  6 ;  the  first 
two,  95,  found  in  the  second  column,  being  common  to 
them  all.  The  logarithm  of  8976  is,  therefore,  3.953083. 

10.  To  denote  the  point  in  which  the  second  figure 
changes,  when  such  change  does  not  take  place  in  the  first 
logarithmic  column,  the  first  of  the  four  figures  from  the 
change  to  the  end  of  the  line  is  printed  as  an  index  figure ; 
thus,  on  page  25  of  the  tables,  we  have  the  lines 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

456 
457 

458 

8965 
9916 
660865 

9060 
°011 
0960 

9155 
°106 
1055 

9250 
°201 
1150 

9346 
°296 
1245 

9441 
°391 
1339 

9536 
°486 
1434 

9631 
°581 
1529 

9726 
°676 
1623 

9821 
°771 
1718 

In  such  cases  the  first  two  figures  are  found  in  the  next 
line.  The  logarithm  of  4575  is,  therefore,  3.660391. 

11.  To  find  the  Logarithm  of  a  number  from  the 
tables.  If  the  number  consists  of  one  or  two  figures  only, 
its  logarithm  is  found  on  the  first  page  of  the  table.  If  the 
two  figures  are  both  integers,  the  index  is  given  also ;  but, 
if  the  one  or  both  figures  be  decimal,  the  decimal  part  only 


22  THE  NATURE  AND  USE   OF  LOGARITHMS.         [CHAP.  I. 

of  the  logarithm  should  be  taken  out.     Thus,  the  loga- 
rithm of  8  is  0.903090 ;  of  59  is  1.770852. 

If  the  number  be  wholly  or  part  a  decimal,  the  index 
must  be  changed  in  accordance  with  the  principles  laid 
down  in  Art.  7.  Thus,  the  index  must  be  one  less  than  the 
number  of  figures  in  the  integral  part  of  the  natural  num- 
ber. But  when  the  natural  number  is  wholly  a  decimal 
the  index  is  negative,  and  must  be  one  more  than  the  num- 
ber of  ciphers  between  the  first  significant  figure  and  tho 
decimal  point.  Thus,  the  logarithm  of 

.8  is  —1.903090 ;  of  .059  is  —2.770852. 

If  the  number  consists  of  three  figures,  look  for  it  in  the 
remaining  pages  of  the  table,  in  the  column  headed  N". 
Opposite  to  it,  in  the  first  column,  will  be  found  the  deci- 
mal portion  of  the  logarithm ;  the  first  two  figures  of  the 
logarithm,  being  common  to  all  the  columns,  are  printed 
but  once,  to  save  room.  Thus,  the  logarithm  of 
272  is  2.434569 ;  of  529  is  2.723456  ; 

the  index  being  placed  in  accordance  with  the  above  rule. 

If  the  number  consists  of  four  figures,  the  first  three 
must  be  found  as  before ;  and  the  fourth,  at  the  top  of  the 
table.  The  last  four  figures  of  the  logarithm  are  found 
opposite  to  the  first  three  figures  of  the  number,  and  under 
the  fourth;  the  first  two  figures  of  the  logarithm  being 
found  in  the  first  logarithmic  column.  Thus,  if  the  num- 
ber were  445.8,  look  for  445  in  the  column  headed  IN",  and 
opposite  thereto,  in  the  column  headed  8,  the  figures  9140 
are  found;  these  affixed  to  64,  found  in  the  first  column, 
give  649140  for  the  decimal  portion  of  the  logarithm ;  and, 
as  there  are  three  integral  figures,  the  index  is  2.  Hence, 
the  complete  logarithm  is  2.649140. 

If  there  are  more  than  four  figures  in  the  number,  find 
the  logarithm  of  the  first* four  figures  as  before.  Take  the 
difference  between  this  logarithm  and  the  next  greater  in 
the  table ;  multiply  this  difference  by  the  remaining  figures 
in  the  number,,  and  from  the  product  separate  as  many 
figures  from  the  right  hand  as  are  contained  in  the  mul- 


SEC.  I.]  THE  NATURE   OF  LOGARITHMS.  23 

tiplier ;  then  add  the  remainder  to  the  logarithm  first  taken 
out :  the  sum  will  he  the  required  logarithm. 
Let  the  logarithm  of  6475.48  be  required. 

The  logarithm  of  6475  is        .811240 
The  next  greater  is  1307 

~67 

67  x  48  =  32,16 

32  added  to  811240  gives  .811272 ; 
and  the  index  being  3,  the  complete  logarithm  is  3.811272. 

Next  let  the  logarithm  of  .0026579  be  required. 

The  logarithm  of  2657  is        .424392 
The  next  greater  4555 

Difference  163 

9 

146,7 

424392  +  147  =  .424539,  and  the  index  being  -3,  the  com- 
plete logarithm  is  —3.424539. 

NOTE. — In  this  last  example,  the  product  is  1467 :  the  figure  stricken  off 
being  7,  which  is  more  than  5,  147  is  taken  instead  of  146. 

EXAMPLES. 
Required  the  logarithms  of  the  following  numbers : — 


1.  Of  7.5  0.875061 

2.  Of  876  2.942504 

3.  Of  93.37  1.970207 

4.  Of  .4725  —1.674402 

5.  Of  .869427  —1.939233 

6.  Of  .01367  —2.135769 


7.  Of  .0645775  —2.810081 

8.  Of  .004679  —3.670153 

9.  Of  37196.2  4.570499 

10.  Of  .14638  —1.165482 

11.  Of  6273.69  3.797523 

12.  Of  .037429  —2.573208 


12.  To  find  the  natural  number  corresponding  to  a 
given  Logarithm.  If  four  figures  only  be  needed  in  the 
answer,  seek  in  the  columns  of  logarithms  for  the  one  near- 
est to  the  decimal  part  of  the  given  logarithm :  the  first 
three  figures  of  the  natural  number  will  be  found  in  the 
column  marked  N ;  and  the  fourth,  at  the  top  of  the  column 
in  which  the  logarithm  is  found. 

When  the  index  is  positive,   the  number  of  integral 


24  THE  NATURE  AND  USE   OF  LOGARITHMS.         [CHAP.  I. 

figures  will  be  one  greater  than  the  number  expressed  by 
the  index ;  but,  if  the  index  is  negative,  the  number  will 
be  wholly  decimal,  and  have  one  less  cipher  between  the 
decimal  point  and  the  first  significant  figure  than  the  num- 
ber expressed  by  the  index.  Thus,  the  natural  number 
corresponding  to  the  logarithm  2.860996  is  726.1;  and  that 
corresponding  to  —2.860996  is  .07261. 

If  the  logarithm  be  found  exactly  in  the  tables,  and  there 
be  not  enough  figures  in  the  corresponding  number,  the 
deficiency  must  be  supplied  by  ciphers.  Thus,  the  natural 
number  corresponding  to  6.891649  is  7792000. 

But,  if  five  or  six  figures  be  required,  find  in  the  table 
the  logarithm  next  less  than  the  given  one,  and  take  out 
the  corresponding  number  as  before;  subtract  this  loga- 
rithm from  the  next  greater  in  the  table,  and  also  from  the 
given  logarithm;  annex  one  or  two  ciphers  to  the  latter 
remainder,  according  as  five  or  six  figures  are  required,  and 
divide  the  result  by  the  former.  The  quotient  annexed  to 
the  figures  first  taken  out  will  give  the  figures  required, 
the  decimal  point  being  placed  as  before. 

Required  the  number  corresponding  to  2.649378,  to  six 
figures 

Given  logarithm  .649378 

Next  less  .649335    cor.  num.  4460 

Difference  43 

Next  greater  logarithm  .649432 

Next  less  .649335 

Difference  ~~97)4300(44 

388 

420 

388 

32 

Hence,  the  number  is  446.044. 

EXAMPLES. 

Required  the  natural  numbers  corresponding  to  the  fol- 
lowing logarithms. 


SEC.  II.] 


ON  THE   USE   OF  LOGARITHMS. 


25 


1.      2.46T415 
2.  —1.396143 
3.     2.041637 
4.  —  3.167149 

Ans.  293.37 
.24897 
110.062 
.0014694 

5.     4.617392 
6.     1.947138 
7.  —2.960014 
8.—  2.760116 

Ans.  41437.3 
88.54 
.    .091204 
.057559 

SECTION  II. 
ON  THE  USE  OF  LOGARITHMS. 

13.  Multiplication.  To  multiply  numbers  by  means  of 
logarithms.  Add  together  the  logarithms  of  the  factors, 
and  take  out  the  natural  number  corresponding  to  the 
sum.  If  any  of  the  indices  be  negative,  the  figure  to  be 
carried  from  the  sum  of  the  decimal  portions  must  be  con- 
sidered positive,  and  added  to  the  sum  of  the  positive,  or 
subtracted  from  the  sum  of  the  negative  indices.  Then 
collect  the  affirmative  indices  into  one  sum,  and  the  nega- 
tive into  another,  take  the  difference  between  these  sums, 
and  prefix  thereto  the  sign  of  the  greater  sum. 

EXAMPLES. 
Ex.  1.  Multiply  47.25  and  397.3. 

log.  1.674402 


47.25 
397.3 
Product,  18772.5 


"    2.599119 
4.273521 


Ex.  2.  Required  the  product  of  764.3,  .8175,  .04729,  and 
.00125. 

log.  2.883264 
"  —1.912488 
«  —2.674769 
«  —3.096910 


764.3 

.8175 

.04729 

.00125 

Product,  .0369344 


—2.567431 


Ex.  3.  Required  the  product  of  87.5  and  6.7. 

Ans.  586.25. 


26  THE  NATURE  AND  USE  OF  LOGAEITHMS.        [CHAP.  I. 

Ex.  4.  Required  the  continued  product  of  .0625,  41.67, 
.81427,  and  2.1463.  Ans.  4.5516. 

Ex.  5.  Multiply  67.594,  .8739,  and  463.92  together. 

Ans.  27404. 
Ex.  6.  Multiply  46.75,  .841,  .037654,  and  .5273  together. 

Ans.  .780633. 

Ex.  7.  Multiply  .00314, 16.2587,  .32734,  .05642,  and  1.7638 
together.  Ans.  .001663. 

14.  Division.  To  divide  numbers  by  logarithms.  Subtract 
the  logarithm  of  the  divisor  from  that  of  the  dividend :  the 
remainder  will  be  the  logarithm  of  the  quotient. 

If  one  or  both  of  the  indices  are  negative,  subtract  the 
decimal  portions  of  the  logarithm  as  before ;  and,  if  there 
be  one  to  carry  from  the  last  figure,  add  it  to  the  index  of 
the  divisor,  if  this  be  positive,  but  subtract  if  it  be  nega- 
tive ;  then  conceive  the  sign  of  the  result  to  be  changed, 
and  if,  when  so  changed,  the*  two  indices  have  the  same 
sign,  add  them  together ;  but,  if  they  have  different  signs, 
take  their  difference  and  prefix  the  sign  of  the  greater. 


EXAMPLES. 

Ex.  1.  Divide        6740        log.      3.828660 
by                87        log.      1.939519 

Quotient,  77.471 

Ex.  2.  Divide       86.47 
by           .0124 
Quotient,  6973.4 

Ex.  3.  Divide        .0642 
by           87.63 
Quotient,  .00073263 

Ex.  4.  Divide        .0642 
by        .008763 

1.889141 

log.      1.936865 
log.  —2.093422 

14.7765 
.012642 

3.843443 

log.  —2.807535 
log.      1.942653 

—4.864882 

log.  —2.807535 
log.  —3.942653 

Quotient,  7.3263                                  0.864882 

Ex.  5.  Divide  407.3  by  27.564.                   Ans. 
Ex.  6.  Divide  .80743  by  63.87.                  Ans. 

SEC.  II.]  ON  THE  USE  OF  LOGARITHMS.  27 

Ex.  7.  Divide  963.7  by  .00416.  Ans.  231659. 

Ex.  8.  Divide  86.39  by  .09427.  Ans.  916.41. 

Ex.  9.  Divide  .006357  by  .0574.  Ans.  .11075. 

Ex.  10.  Divide  76.342  by  .09427.  Ans.  809.82. 

15.  To  involve  a  number  to  a  power.    Multiply  the 
logarithm  of  the  number  by  the  index  of  the  power  to 
which  it  is  to  be  raised. 

If  the  index  of  the  logarithm  is  negative,  and  there  is 
any  thing  to  be  carried  from  the  product  of  the  decimal 
part  by  the  multiplier,  instead  of  adding  this  to  the  pro- 
duct of  the  index,  subtract  it:  the  difference  will  be  the 
index  of  the  product,  and  will  always  be  negative. 

Ex.  1.  Required  the  fourth  power  of  5.5. 

5.5                       log.  0.740363 
4 

915.065  2.961452. 

Ex.  2.  Required  the  fifth  power  of  .63. 

.63  log.  —1.799341 

5 

.099244  —2.996705. 

Ex.  3.  Required  the  fourth  power  of  7.639. 

Ans.  3405.24. 

Ex.  4.  Required  the  third  power  of  .03275. 

Ans.  .00003513. 

Ex.  5.  "What  is  the  fifteenth  power  of  1.06  ? 

Ans.  2.3966. 

Ex.  6.  What  is  the  sixth  power  of  .1362  ? 

Ans.  .0000063836. 

Ex.  7.  What  is  the  tenth  power  of  .9637  ? 

Ans.  .69091. 

16.  To  extract  a  given  root  of  a  number.    Divide  the 
logarithm  of  the  number  by  the  degree  of  the  root  to  be 
extracted :  the  quotient  will  be  the  logarithm  of  the  root. 

If  the  index  of  the  logarithm  is  negative,  and  does  not 


28  THE  NATURE  AND  USE  OF  LOGARITHMS.        [CHAP.  I. 

contain  the  divisor  an  exact  number  of  times,  increase  it 
by  so  many  as  are  necessary  to  make  it  do  so,  and  carry 
the  number  so  borrowed,  as  so  many  tens  to  the  first  figure 
of  the  decimal. 

Ex.  1.  Extract  the  fourth  root  of  56.372. 
56.372  log.  4)1.751063 

ftesult,  2.7401  .437766 

Ex.  2.  Extract  the  fifth  root  of  .000763. 

.000763  log.  5)— 4.882525 

Result,  .23796  —1.376505. 

Ex.  3.  What  is  the  fifth  root  of  .00417  ?  Ans.  .3342. 

Ex.  4.  Required  the  fourth  root  of  .419.  Ans.  .80455. 
Ex.  5.  Required  the  tenth  root  of  8764.5.  Ans.  2.479. 
Ex.  6.  Required  the  seventh  root  of  .046375. 

Ans.  .6449. 

Ex.  7.  Required  the  fifth  root  of  .84392.  Ans.  .96663. 
Ex.  8.  Required  the  sixth  root  of  .0043667.  Ans.  .40429. 

17.  Arithmetical  Complements.  When  several  num- 
bers are  to  be  added,  and  others  subtracted  from  the  sum, 
it  is  often  more  convenient  to  perform  the  operation  as 
though  it  were  a  simple  case  of  addition.  This  may  be 
done  by  conceiving  each  subtractive  quantity  to  be  taken 
from  a  unit  of  the  next  higher  order  than  any  to  be  found 
among  the  numbers  employed ;  then  add  the  results  with 
the  additive  numbers,  and  deduct  from  the  result  as  many 
units  of  the  order  mentioned  as  there  were  subtractive 
numbers.  The  difference  between  any  number  and  a  unit 
of  the  next  higher  order  than  the  highest  it  contains  is 
called  the  arithmetical  complement  of  the  number.  Thus,  the 
arithmetical  complement  of  8765  is  1235.  It  is  easily  ob- 
tained by  taking  the  first  significant  figure  on  the  right  from 
ten,  and  each  of  the  others  from  nine.  This  may  be  done 
mentally,  so  that  the  arithmetical  complements  need  not  be 
written  down. 

Thus,  suppose  A  started  out  with  375  dollars  to  collect 


SEC.  II.] 


ON  THE   USE   OF   LOGARITHMS. 


29 


some  bills  and  to  pay  sundry  debts.  From  B  he  received 
$104,  to  D  he  pays  $215,  to  E  he  pays  $75,  from  F  he  re- 
ceives $437,  and,  finally,  pays  to  Gr  $137.  How  much  has 
he  left? 

375 
.  104 

which  are  added  as 
though  they  were 


375' 

104 

--215 

—  75 

437 

—137 

489 


785 
925 
437 
863 


3489, 


deducting  3000  from  the  final  result  3489,  because  there 
were  three  subtractive  quantities. 

The  arithmetical  complements  of  logarithms  are  gene- 
rally employed  where  there  are  more  subtractive  logarithms 
than  one.  To  give  symmetry  to  the  result,  it  would  be 
neater  to  employ  them  in  all  cases.  To  a  person  who  has 
much  facility  in  calculation,  it  is  most  convenient  to  write 
down  the  logarithm  as  taken  from  the  table,  and  obtain 
the  arithmetical  complement  as  the  work  is  carried  on. 
Thus,  in  the  example  above,  the  numbers  could  be  written 
as  in  the  first  column ;  but  in  the  addition,  instead  of  em- 
ploying the  figures  as  they  appear  in  the  subtractive  num- 
ber, the  complement  of  the  first  significant  figure  to  ten, 
and  of  the  others  to  nine,  should  be  employed. 

As  an  example  of  the  use  of  the  arithmetical  comple- 
ments of  the  logarithms  of  numbers,  let  it  be  required  to 

27    475 

work  by  logarithms  the  proportion  as  —  :  -^=-  : :  125  :  x. 

oo      17 

Here,  as  the  first  term  is  a  fraction,  it  will  have  to  be  in- 
verted ;  and  the  question  will  be  the  same  as  finding  the 

„  55  x  475  x  125 

value  of 


27  x  l\ 


log. 
u 


27 
17 
«  55 
«  475 
«  125 


1.431364^  which  are 
1.230449      added  as 
1.740363  V   though 
2.676694    they  were 
2.096910 j    written 

Result,  7114.66    3.852154 


A.  C.  8.568636 

A.  C.  8.769551 

1.740363 

2.676694 

2.096910 

3.852154T 


30  THE  NATURE  AND  USE  OP  LOGARITHMS.        [CHAP.  I. 

deducting  20,  because  there  were  two  arithmetical  comple- 
ments employed. 

In  the  examples  wrought  out  in  the  subsequent  part  of 
this  work,  the  arithmetical  complements  of  the  logarithms 
of  the  first  term  of  every  proportion  are  employed. 


CHAPTER  II. 

PRACTICAL    GEOMETRY. 

SECTION  I. 
DEFINITIONS, 

18.  THE  practical  surveyor  will  find  a  good  knowledge 
of  Algebra  and  of  the  Elements  of  Geometry  an  invaluable 
aid  not  only  in  elucidating  the  principles  of  the  science, 
but  in  enabling  him  to  overcome  difficulties  with  which  he 
will  be  certain  to  meet.     In  fact,  so  completely  is  Survey- 
ing dependent  on  geometrical  principles,  that  no  one  can 
obtain  other  than  a  mere  practical  knowledge  of  it,  without 
first  having  mastered  them;  and  he  who  depends  solely 
on  his  practical  experience  will  be  certain  to  meet  with 
cases  which  will  call  for  a  kind  of  knowledge  which  he 
does  not  possess,   and  which  he   can   obtain   only   from 
Geometry. 

Every  student,  therefore,  who  desires  to  become  an  in- 
telligent surveyor,  should  first  study  Euclid,  or  some  other 
treatise  on  Geometry.  He  will  then  have  a  key  which  will 
not  only  unlock  the  mysteries  contained  in  the  ordinary 
practice,  but  which  will  also  open  the  way  to  the  solution 
of  all  the  more  difficult  cases  which  occur.  To  those  who 
have  taken  the  course  above  recommended,  the  problems 
solved  in  the  present  chapter  will  be  familiar.  They  are 
inserted  for  the  benefit  of  those  who  may  not  be  thus  pre- 
pared, and  also  as  affording  some  of  the  most  convenient 
modes  of  performing  the  operations  on  the  ground. 

19.  Geometry  is  the  science  of  magnitude  and  position. 

31 


32  PRACTICAL  GEOMETRY.  [CHAP.  II. 

20.  A  solid  is  a  magnitude  having  length,  breadth,  and 
thickness. 

All  material  bodies  are  solids,  and  so  are  all  portions  of 
space,  whether  they  are  occupied  with  material  substances 
or  not.  Geometry,  treating  only  of  dimension  and  posi- 
tion, has  no  reference  to  the  physical  properties  of  matter. 

21.  The  surfaces  of  solids  are  superficies.    A  superficies 
has,  therefore,  only  length  and  breadth. 

22.  The  boundaries  of  superficies,  and  the  intersection 
of  superficies,  are  lines.     Hence,  a  line  has  length  only. 

23.  The  extremities  of  lines,  and  the  intersections  of 
lines,  are  points.    A  point  has,  therefore,  neither  length, 
breadth,  or  thickness. 

24.  A  pointy  therefore,  may  be  defined  as  that  which  has 
position,  but  not  magnitude. 

25.  A  line  is  that  which  has  length  only. 

26.  A  straight  line  is  one  the  direction  of  which  does  not 
change.     It  is  the  shortest  line  that  can  be  drawn  between 
two  points. 

27.  A  superficies  has  length  and  breadth  only. 

28.  A  plane  superficies,  generally  called  simply  a  plane,  is 
one  with  which  a  straight  line  may  be  made  to  coincide  in 
any  direction. 


29.  A  plane  rectilineal  angle,  or  sim- 
ply an  angle,  is  the  inclination  of 
two  lines  which  meet  each  other. 
(Fig.  1.) 

A. 


30.  An  angle  may  be  read  either  by  the  single  letter  at 


SEC.  L]  DEFINITIONS.  33 

the  intersection  of  the  lines,  or  by  three  letters,  of  which 
that  at  the  intersection  must  always  occupy  the  middle. 
Thus,  (Fig.  1,)  the  angle  between  BA  and  AC  may  be  read 
simply  A  or  BAG. 

31.  The  magnitude  of  an  angle  has  no  reference  to  the 
space  included  between  the  lines,  nor  to  their  length,  but 
solely  to  their  inclination. 

32.  Where  one  straight  line  stands  on  another  so  as  to 
make  the  adjacent  angles  equal, 

each  of  these  angles  is  called  a 
right  angle;  and  the  lines  are  said 
to  be  perpendicular  to  each  other. 
Thus,  (Fig.  2,)  if  ACD  =  BCD, 

each  is  a  right  angle,  and  CD  is     

perpendicular  to  AB.  A  c  B 

33.  An  angle  less  than  a  right  angle  is  called  an  acute 
angle.     Thus,  BCE  or  ECD  (Fig.  2)  is  an  acute  angle. 

34.  An  angle  greater  than  a  right  angle  is  called  an 
obtuse  angle.    ACE  (Fig.  2)  is  an  obtuse  angle. 

35.  The  distance  of  a  point  from  a  straight  line  is  the 
length  of  the  perpendicular  from  that  point  to  the  line. 

36.  Parallel  straight  lines  are  those  of  which  all  points 
in  the  one  are  equidistant  from  the  other. 

37.  A.  figure  is  an  enclosed  space. 

38.  A  triangle  is  a  figure  bounded  by  three  straight  lines. 

39.  An  equilateral  triangle  is  one  the  three  sides  of  which, 
are  equal. 

40.  An  isosceles  triangle  is  one  of  which  two  of  the  sides 
are  equal.    The  third  side  is  called  the  base. 


34  PRACTICAL  GEOMETRY.  [CHAP.  II. 

4L  A  scalene  triangle  has  three  unequal  sides. 

42.  A  right-angled  triangle  has  one  of  its  angles  a  right 
angle. 

43.  The  side  opposite  the  right  angle  is  called  the  hypo* 
thenuse,  and  the  other  sides,  the  legs. 

44.  An  obtuse-angled  triangle  has  one  of  its  angles  obtuse. 

45.  A  quadrilateral  figure  is  bounded  by  four  sides. 

Fig.  3. 

46.  A  parallelogram  (Fig.  3)  is  a 
quadrilateral,  the  opposite  sides  of 
which  are  parallel. 


47.  A  rectangle  (Fig.  4)  is  a  parallelogram,  the  adjacent 
sides  of  which  are  perpendicular  to  each  Fig.  4. 
other.     Thus,  ABCD  is  a  rectangle.     A  A 

rectangle  is  read  either  by  naming  the 
letters  around  it  in  their  order,  or  by 
naming  two  of  the  sides  adjacent  to  any 
angle.  Thus,  the  rectangle  ABCD  is 
read  the  rectangle  AB.BC. 

Whenever  the  rectangle  of  two  lines,  such  as  DE.EF,  is 
spoken  of,  a  rectangular  parallelogram,  the  adjacent  sides 
of  which  are  equal  to  the  lines  DE  and  EF,  is  meant. 

48.  A  square  is  a  rectangle,  all  the  sides  of  which  are 
equal. 

49.  A  rhombus  is  an  oblique  parallelogram,  the  sides  of 
which  are  equal. 

50.  A  rhomboid  is  an  oblique  parallelogram,  the  adjacent 
sides  of  which  are  unequal. 


SEC.  I.] 


DEFINITIONS. 


35 


51.  All  quadrilaterals   that  are  not  parallelogiams  are 
called  trapeziums. 

52.  A  trapezoid  is  a  trapezium,  having  two  of  its  sides 
parallel. 

53.  Figures  of  any  number  of  sides  are  called  polygons, 
though  this  term  is  generally  restricted  to  those  having 
more  than  four  sides. 


54.  The  diagonal  of  a  figure  is  a  line  joining  any  two 
opposite  angles. 


55.  The  base  of  any  figure  is  the 
side  on  which  it  may  be  supposed 
to  stand.  Thus,  AB  (Fig.  5)  is  the 
base  of  ABCD. 


Fig.  5. 


56.  The  altitude  of  a  figure  is  the  distance  of  the  highest 
point  from  the  line  of  the  base.     CE  (Fig.  5)  is  the  altitude 
of  ABCD. 

57.  The  diameter  of  a  circle  is  a  straight  line  through  the 
centre,  terminating  in  the  circumference. 

58.  The  radius  of  a  circle  is  a  straight  line  drawn  from 
the  centre  to  the  circumference. 


Fig.  6. 


59.  A  segment  of  a  circle  is  any  part 
cut  off  by  a  straight  line.  Thus, 
ABCD  is  a  segment. 


36 


PRACTICAL   GEOMETRY. 


[CHAP.  II. 


60.  A  semicircle  is  a  segment  cut  off 
by  the  diameter.  ABCandAEB  (Fig. 
7)  are  semicircles. 


61.  A  quadrant  is  a  portion  of  a  circle  included  between 
two  radii  at  right  angles  to  each  other.     ADCandBDC 
(Fig.  7)  are  quadrants. 

62.  The  angle  in  a  segment  is  the  angle  contained  between 
two  straight  lines  drawn  from  any  point  in  the  arc  of  a  seg- 
ment to  the  extremities  of  that  arc.     Thus,  ABD  and  ACD 
(Fig.  6)  are  angles  in  the  segment  ABCD. 

63.  Similar  rectilineal  figures  have  their  angles  equal, 
and  the  sides  about  the  equal  angles  proportionals. 

64.  Similar  segments  of  a  circle  are  those  which  contain 
equal  angles. 


SECTION  II. 

GEOMETRICAL  PROPERTIES  AND  PROBLEMS. 

A.— GEOMETRICAL  PROPERTIES. 

65.  ALL  right  angles  are  equal  to  each  other. 

66.  The  angles  which  one  straight  line  makes  with  an- 
other on  one  side  of  it  are  together  equal  to  two  right 
angles.     Thus,  ACE  and  ECB  (Fig.  2)  are  together  equal  to 
two  right  angles.     (13.1.) 


SEC.  II.]   GEOMETRICAL  PROPERTIES  AND  PROBLEMS.       37 

67-  If  a  number  of  straight  lines  are  drawn  from  a  point 
in  another  straight  line,  all  the  successive  angles  are  together 
equal  to  two  right  angles.  Thus,  A  CD  +  DCE  +  ECB  (Fig. 
2)  make  two  right  angles. 

Fig.  8. 

A 

68.  If  two  straight  lines  inter- 
sect each  other,  the  angles  verti- 
cally opposite  are  equal.     Thus, 
AEC  (Fig.  8)  =  BED,  and  AED  = 
BEG.    (15.1.) 

69.  Triangles  which  have  two  sides  and  the  included 
angle  of  one  respectively  equal  to  the  two  sides  and  the 
included  angle  of  the  other,  are  equal  in  all  respects.    (4.1.) 

70.  Triangles  which  have  two  angles  and  the  interjacent 
side  of  one  respectively  equal  to  two  angles  and  the  inter- 
jacent side  of  the  other,  are  equal  in  all  respects.     (26.1.) 

71.  Triangles  which,  have  two  angles  of  the  one  respec- 
tively equal  to  two  angles  of  the  other,  and  which  have  also 
the  sides  opposite  to  two  equal  angles  equal  to  each  other, 
are  equal  in  all  respects.     (26.1.) 

72.  If  a  straight  line  cuts  two  pa-  Fig.  9- 
rallel  lines,  the  angles  similarly  situ- 
ated in  respect  to  these  lines,  and              \ 

also  those  alternately  situated,  will  be  A -^ B 

equal  to  each  other  (29.1.)     Thus,  \ 

(Fig.  9,)  EFB  =  FGD,  BFG  =  DGS,  \ G 

AFE  =  CGF,  and  AFG  =  CGH,  c V~ 

being  similarly  situated  ;  and  AFE  '  \ 

=  DGH,  EFB  =  CGH,  AFG  =  k 

FGD,  and  BFG  =  FGC,  being  alternately  situated. 

73.  If  a  straight  line  cuts  two  parallel  straight  lines,  the 
two  exterior  angles  on  the  same  side  of  the  cutting  line, 
and  also  the  two  interior  angles,  are  equal  to  two  right 


38  PRACTICAL  GEOMETRY.  [CHAP.  II. 

angles.  Thus,  (Fig.  9,)  EFB  and  DGH  are  equal  to  two 
right  angles,  as  are  also  AFE  and  CGH.  So  also  the  pairs 
of  interior  angles  AFG  and  FGC,  BFG  and  FGD,  are  each 
equal  to  two  right  angles.  (29.1.) 

74.  The  angles  at  the  base  of  an  isosceles  triangle  are 
equal  to  each  other.     (5.1.) 

75.  If  one  side  of  a  triangle  be 
produced,    the     exterior    angle    so 
formed  will   be   equal  to  the  two 
angles  adjacent  to  the  opposite  side, 
and  the  three  interior   angles   are 
equal  to  two   right  angles.     Thus, 
(Fig.  10,)  ACD  =  ABC  +  BAG,  and    B 

ABC  +  BAC  +  ACB  =  two  right  angles.     (32.1.) 

76.  The  interior  angles  of  any  rectilineal  figure  are  equal 
to  twice  as  many  right  angles  as  the  figure  has  sides,  dimi- 
nished by  four  right  angles.     The  interior  angles  of  a  quadri- 
lateral are  therefore  equal  to  four  right  angles.     (Cor.  1, 
32.1.) 

77.  The  opposite  sides  and  angles  of  a  parallelogram  are 
equal  to  each  other.     (34.1.) 

78.  Conversely,  any  quadrilateral  of  which  the  opposite 
sides  or  the  opposite  angles  are  equal  is  a  parallelogram. 

79.  Parallelograms  having  equal  bases  and  altitudes,  and 
also  triangles  having  equal  bases  and  altitudes,  are  equal  to 
each  other.     (35-38.1.) 

80.  A  parallelogram  is  double  a  triangle  having  the  same 
base  and  altitude.     (41.1.) 

81.  The  square  on  the  hypothenuse  of  a  right-angled 
triangle  is  equal  to  the  sum  of  the  squares  of  the  legs. 
(47.1.) 


SEC.  II.]   GEOMETRICAL  PROPERTIES  AND  PROBLEMS.       39 

82.  Any  figure  described  on  the  hypothenuse  of  a  right- 
angled  triangle  is  equal  to  the  sum  of  the  similar  figures 
similarly  described  on  the  sides.  (31.6.) 


Fig.  11. 


83,  The  angle  at  the  centre  of  a 
circle  is  double  the  angle  at  the  cir- 
cumference on  the  same  base.  Thus, 
the  angle  at  C  (Fig.  11)  is  double 
either  D  or  E.  (20.3.) 


84.  Angles  in  the  same  segment  of  a  circle  are  equal. 
Thus,  D  and  E  (Fig.  11)  are  equal. 

85.  The  angle  in  a  semicircle  is  a  right  angle ;  the  angle 
in  a  segment  greater  than  a  semicircle  is  acute ;  and  that  in 
a  segment  less  than  a  semicircle  is  obtuse. 

86.  The  sides  about  the  equal  angles  of  equiangular  tri- 
angles are  proportional.     (4.6.) 

B.— GEOMETRICAL  PROBLEMS. 

Under  this  head  are  given  those  methods  of  construction 
which  are  applicable  to  paper  drawings.  The  methods  to 
be  used  in  field  operations  will  be  given  in  a  subsequent 
chapter. 

Fig.  12. 

87.  Problem  1. —  To  bisect  a  given 
straight  line.     Let  AB  (Fig.  12)  be  the 
given  line.     "With  the  centres  A  and 

B,  and  radius  greater  than  half  AB,  iB 


A \ 

describe  arcs  cutting  in  C  and  D. 
Join  CD  cutting  AB  in  E,  and  the 
thing  is  done.  (10.1.) 


40 


PRACTICAL  GEOMETRY. 


[CHAP.  II. 


Problem  2.  To  draw  a  perpendicular  to  a  straight  line  from 
a  given  point  in  it. 
a.  When  the  point  is  not  near  the. end. 


88.  Let  AB  (Fig.  13)  be  the  line  and  0 
the  given  point.  Lay  off  CD  =  CE,  and 
with  D  and  E  as  centres,  and  any  radius 
greater  than  DC,  describe  arcs  cutting  in 
F.  Draw  CF,  and  the  thing  is  done. 
(11.1) 


Fig.  13. 


b.  When  the  point  is  near  the  end  of  the  line. 

89.  First  Method. — Take  any  point 
D  (Fig.  14)  not  in  the  line,  and  with 
the  centre  D  and  radius  DC  de- 
scribe the  circle  ECF,  cutting  AB  in 
E.  Join  ED  and  produce  it  to  F. 
Then  will  CF  be  the  perpendicular. 
For  ECF,  being  an  angle  in  a  semi-  A~~ 
circle,  is  a  right  angle.  (85.) 


90.  Second  Method.— With  C 
(Fig.  15)  and  any  radius  describe 
DEF ;  with  D  and  the  same  radius 
cross  the  circle  in  E ;  and  with  E 
as  a  centre,  and  the  same  radius, 
cross  it  in  F.  "With  E  and  F  as 
centres,  and  any  radius,  describe 
arcs  cutting  in  G.  Then  will  C  Gi- 
be the  perpendicular. 


Fig.  14. 


Fig.  15. 


C     B 


Problem  3. — To  let  fall  a  perpendicular  to  a  line  from  a  point 
without  it. 
a.  When  the  point  is  not  nearly  opposite  the  end  of  the  line. 


SEC.  II.]   GEOMETRICAL  PROPERTIES  AND  PROBLEMS. 


41 


Fig.  16. 
C 


91.  Let  AB  (Fig.  16)  be  the  line 
and  C  the  given  point.  With  the 
centre  C  describe  an  arc  cutting  AB 
in  D  and  E.  With  the  centres  D  and 
E  and  any  radius  describe  arcs  cut- 
ting in  F.  Join  CF,  and  the  thing 
is  done.  (12.1.) 


b.  When  the  point  is  nearly  opposite  the  end  of  the  line. 

Fig.  17. 

92.  First  Method.—With  D  and  E 
as  centres,  and  radii  DC  and  EC,  de- 
scribe arcs  cutting  in  F :  then  will  CF 
be  the  perpendicular.  For,  the  tri- 
angles CDE  and  FDE  being  equal, 
(8.1,)  DGC  and  FGD  will  be  equal. 
(4.1.) 


X'  '/ 

\" 
\ 
\ 

"\.\ 

*l 

I 

93.  Second  Method.— Let  F  (Fig.  14)  be  the  point.    From 
F  to  any  point  E  in  the  line  AB  draw  FE.     On  it  describe 
a  semicircle  cutting  AB  in  C.    Join  F  and  C,  and  FC  will 
be  the  perpendicular    (85.) 

Problem  4. — At  a  given  point  in  a  given  straight  line  to 
make  an  angle  equal  to  a  given  angle. 

94.  Let  BCD  (Fig.  18)  be  the  given 
angle,,  and  A  the  given  point  in  AE. 
With  the  centre  C  and  any  radius  de- 
scribe BD,  cutting  the  sides  of  the  angle 
in  B  and  D.     With  A  as  a  centre  and 
the  same  radius  describe  EF ;  make  EF 
=  DB ;  draw  AF,  and  the  thing  is  done. 


PRACTICAL  GEOMETRY. 


[CHAP.  II. 


Problem  5. —  To  bisect  a  given  angle. 

95.  Let  BAG  (Fig.  19)  be  the  given 
angle.  With  the  centre  A  and  any  radius 
describe  an  arc  cutting  the  sides  in  B  and  C. 
"With  the  centres  B  and  C,  and  the  same  or 
any  other  radius,  describe  arcs  cutting  in 
D.  Join  AD,  and  the  thing  is  done.  (9.1.) 


Fig.  20. 


Problem  6. — To  draw  a  straight  line  touching  a  circle  from 
a  given  point  without  it. 

96.  Let  ABC  be  the  given 
circle,  and  D  the  given  point. 
Join  D  and  the  centre  E.     On 
DE  describe  a  semicircle  cut- 
ting the   circumference  in  B. 
Join  DB,  and  it  will  be  the  tan- 
gent required. 

For  DBE,  being  an  angle  in  a  semicircle,  is  a  right  angle, 
(31.3 ;)  therefore,  DB  touches  the  circle,  (16.3.) 

If  the  point  were  in  the  circumference  at  B.  Join  EB, 
and  draw  BD  perpendicular  to  it.  BD  will  be  the  tangent. 

Problem  7. —  Through  a  given  point  to  draw  a  line  parallel 
to  a  given  straight  line. 

97.  First  Method.— Let  A  (Fig.  21)        p       *ig.  21.    A 
be  the  given  point,  and  BC  the  given    - 

line.    From  A  to  BC  let  fall  a  per- 
pendicular AD;    and  at  any   other 
point  E  in  BC  erect  a  perpendicular  B    E 
EF  equal  to  AD.     Through  A  and  F  draw  AF,  which  will 
be  the  parallel  required. 


98.  Second  Method. — From  A  (Fig. 
22)  to  D,  any  point  in  BC,  draw  AD. 
Make  DAE  =  ADC,  and  AE  will  be 
parallel  to  BC.  (2T.1.) 


Fig.  22. 


B    D 


SEC.  II.]        GEOMETRICAL  PROPERTIES  AND  PROBLEMS. 


43 


99.  Third  Method.— Through  A  draw 
ADE,  cutting  BC  in  D.  Make  DE  = 
AD.  Through  E  draw  any  other  line 
EFG,  cutting  BC  in  F.  Make  FG  = 
EF :  then  AG  will  be  parallel  to  BC. 
(2.6.) 


Fig.  23. 

E 


\ 


Problem  8. — To  inscribe  a  circle  in  a  given  triangle. 

Fig.  24. 

100.  Let  ABC  (Fig.  24)  be  the 
given  triangle.  Bisect  two  of  its 
angles  A  and  B  by  the  lines  AD, 
BD,  cutting  in  D.  Then  will  D  be 
the  centre.  (4.4.) 


Problem  9. — To  describe  a  circle  about  a  given  triangle. 

Fig.  25. 

101.  Bisect  two  of  the  sides,  as  AC 
and  AB,  (Fig.  25,)  by  the  perpendicu- 
lars FE  and  DE,  cutting  in  E.  Then 
will  E  be  the  centre  of  the  required 
circle. 


Problem  10. — To  find  a  third  proportional  to  two  straight 
lines. 


Fig.  26. 


102,  Let  M  and  K"  (Fig.  26)  be 
the  given  lines.  Draw  two  lines 
AB  and  AC,  making  any  angle  at 
A.  Lay  off  AD  =  M,  and  AE  and 
AF  each  equal  to  K  Join  DF, 
and  draw  EG  parallel  to  it.  AG  A 
will  be  the  third  proportional  re- 
quired. (11.6.) 


Problem  11. — To  find  a  fourth  proportional  to  three  given 
straight  lines. 


44 


PRACTICAL   GEOMETRY. 


[CHAP.  IL 


103.  Let  M,  1ST,  and  0  (Fig.  27) 
be  the  three  lines.     Draw  any  two 
lines  AB  and  AC,  meeting  at  A. 
Lay  off  AD  =  M,  AE  =  N,  and  AF 
=  0.    Join  DF,  and  draw  EG  pa-    A 
rallel  to  it :  then  AG  is  the  fourth   M 
proportional  required.     (12.6.)  N 


Fig.  27. 


Problem  12. —  To  find  a  mean  proportional  between  two 
straight  lines. 

104.  First  Method.— Place  the  lines  Fig.  28. 
AB  and  BC    (Fig.  28)  in   the   same 

straight  line.  On  AC  describe  a 
semicircle  cutting  the  perpendicular 
through  B  in  D.  BD  will  be  the 
mean  proportional  required.  (13.6.) 

105.  Second  Method.— Let  AB  and  Fig.  29. 
AC  (Fig.  29)  be  the  given  lines.     On 

AB  describe  a  semicircle  cutting  the 
perpendicular  at  C  in  D.  Join  AD. 
AD  is  the  mean  proportional  required. 
(Cor.  8.6.)  MakeAE  =  AD. 

NOTE. — This  is  a  very  convenient  construction,  and  is  often  employed  in  the 
Division  of  Land. 


C     E 


Fig.  30. 


Problem  13. — To  divide  a  given  line  into  parts  having  the 
same  ratio  as  two  given  numbers  M  and 

106.  Let  AB  (Fig.  30)  be  the  given 
line.  Draw  AC  making  any  angle 
with  AB.  Lay  off  AD  =  M,  taken 
from  any  scale  of  equal  parts,  and 
DE  =  N",  taken  from  the  same  scale. 
Join  BE,  and  draw  DF  parallel  to  it, 
and  the  thing  is  done.  (2.6.) 


CHAPTER  III. 

PLANE    TRIGONOMETRY. 


SECTION  I. 
DEFINITIONS, 

107.  PLANE  TRIGONOMETRY  is  the  science  which  treats  of 
the  relations  between  the  sides  and  angles  of  plane  tri- 
angles;   which  develops  the  principles  by  which,  when 
any  three  of  the  six  parts  of  a  triangle, — viz. :  the  three 
angles  and  the  three  sides, — except  the  three  angles,  are 
given,  the  others  may  be  found'.     It  likewise  treats  of  the 
properties  of  the  trigonometrical  functions. 

108.  Measure  of  Angles.    An  angle  is  the  inclination 
between  two  straight  lines:  it  is  measured  by  the  inter- 
cepted arc  of  a  circle  described  about  the  angular  point  as 
a  centre. 

In  the  measurement  of  angles,  it  is  not  the  absolute 
length  of  the  arc  that  is  needed,  but  the  ratio  which  that 
length  bears  to  the  whole  circumference. 

For  the  purpose  of  expressing  this  ratio  readily,  the  cir- 
cumference is  supposed  to  be  divided  into  360  parts,  called 
degrees,  each  degree  into  60  parts,  called  minutes,  and 
each  minute  into  60  seconds.  Degrees  are  marked  with  a 
cipher  °  over  them,  minutes  with  one  accent ',  and  seconds 
with  two  ".  Thus,  37  degrees,  45  minutes,  and  30  seconds, 
would  be  written  37°  45'  30". 

When  we  speak  of  an  arc  of  35°,  we  mean  an  arc  which 

or 

is  -$^.  of  the  circumference.     An  arc  of  180°  is  half  the 
obU 

45 


46 


PLANE   TRIGONOMETRY. 


[CHAP.  III. 


circumference,  one  of  90°  is  a  quadrant,  and  of  45°  the 
half  of  a  quadrant. 

It  is  evident  that,  if  several  circles  be  described  about 
the  same  point,  the  arcs  intercepted  between  two  lines 
drawn  from  the  centre  will  bear  the  same  ratio  to  the  cir- 
cumferences of  which  they  are  portions.  Thus,  if  around 


Fig.  31. 


the  point  A  (Fig.  31)  two  circles 
BCD  and  EFG  be  described,  cut- 
ting AK  and  AH  in  B,  E,  C,  F, 
the  arc  BC  will  have  to  the  cir- 
cumference BCD  the  same  ratio 
as  EF  has  to  the  circumference 
EFGr.  In  the  measurement  of 
angles,  it  is  a  matter  of  indif- 
ference, therefore,  what  radius  is 
assumed  as  that  of  the  circle  of  reference.  The  radius 
which  is  generally  adopted  is  unity.  This  value  of  the 
radius  makes  it  unnecessary  to  write  it  down  in  the 
formulse. 

The  radius  adopted  in  tfre  construction  of  the  Table  of 
Logarithmic  Sines  and  Tangents,  to  be  described  hereafter, 
is  10,000,000,000. 


Fig.  32. 


109.  The  complement  of  an  arc  or 
angle  is  what  it  differs  from  a  quad- 
rant, or  90°.  Thus,  DB  (Fig.  32)  is 
the  complement  of  AB,  and  MD  of 
AM. 


110.  The  supplement  of  an  arc  or  angle  is  what  it  wants 
of  180°.     Thus,  BE  (Fig.  32)  is  the  supplement  of  AB,  and 
ME  of  AM. 

111.  Trigonometrical  Functions.     The  trigonometri- 
cal functions  are  lines  having  definite  geometrical  relations 
to  the  arc  to  which  they  belcng.     Those  most  in  use  are 
the  sine,  the  cosine,  the  tangent,  the  cotangent,  the  secant, 
and  the  cosecant. 


SEC.  I.]  DEFINITIONS.  47 

The  chord  of  an  arc  is  the  right  line  joining  the  extremi- 
ties of  that  arc.  Thus,  EM  (Fig.  32)  is  the  chord  of  the 
arc  EM. 

The  sine  of  an  arc  is  the  line  drawn  from  one  extremity 
of  the  arc,  perpendicular  to  the  diameter  through  the  other 
extremity.  BF  (Fig.  32)  is  the  sine  of  AB  or  of  EB,  and 
BL  of  BD. 

NOTE. — The  sine  of  an  arc  is  equal  to  the  sine  of  its  supplement. 

The  cosine  of  an  arc  is  the  line  intercepted  between  the 
foot  of  the  sine  and  the  centre.  CF  is  the  cosine  of  AB 
or  of  BE. 

Since  CF  =  BL,  it  is  manifest  that  the  cosine  of  an  arc 
is  equal  to  the  sine  of  its  complement. 

The  tangent  of  an  arc  is  a  line  touching  the  arc  at  one 
extremity  and  produced  till  it  meets  the  radius  through 
the  other  extremity.  Thus,  AT  is  the  tangent  of  AB,  and 
DK  of  DB. 

The  cotangent  of  an  arc  is  the  tangent  of  its  complement. 
Thus,  DK  (Fig.  32)  is  the  cotangent  of  AB. 

The  secant  of  an  arc  is  the  line  intercepted  between 
the  centre  and  the  extremity  of  the  tangent.  Thus,  CT 
(Fig.  32)  is  the  secant  of  AB. 

The  cosecant  of  an  arc  is  the  secant  of  the  complement 
of  that  arc.  Thus,  CK  (Fig.  32)  is  the  cosecant  of  AB. 

The  sine,  cosine,  &c.  of  an  arc  are  also  called  the  sine, 
cosine,  &c.  of  the  angle  measured  by  that  arc.  Thus,  BF 
and  CF  (Fig.  32)  are  the  sine  and  cosine  of  the  angle  ACB. 

NOTE. — The  tangent,  cotangent,  secant,  or  cosecant  of  an  arc  is  equal  to  the 
tangent,  cotangent,  secant,  or  cosecant  of  its  supplement. 

112.  Properties  of  the  Sines,  Tangents,  &c.  of  an 
arc  or  angle. 

The  sine  of  90°,  the  cosine  of  0°,  the  tangent  of  45°,  the 
cotangent  of  45°,  the  secant  of  0°,  and  the  cosecant  of  90°, 
are  each  equal  to  radius. 

The  square  of  the  sine  -f  the  square  of  the  cosine  of 


48  PLANE  TRIGONOMETRY.  [CHAP.  III. 

any  arc  is  equal  to  the  square  of  radius.  (Sin.2  a  +  cos.2  a 
—  Ra.)  This  is  evident  from  the  right-angled  triangle  CFB, 
(Fig.  32.)  (4T.1.) 

The  square  of  the  tangent  -f  the  square  of  radius  is  equal 
to  the  square  of  the  secant.  Tan.2  a  -f  E2  =  sec.2  a.  (47.1.) 

Tan.  a  :  R  : :  K  :  cotan.  a,  or  tan.  a.  cot.  a  —  R2.  This  is 
evident  from  the  similarity  of  the  triangles  ACT  and  DKC, 
(Fig.  32,)  which  give  (4.6)  AT  :  AC  : :  CD  :  DK 

The  sine  of  30°  and  the  cosine  of  60°  is  each  equal  to 
half  radius. 

113.  Geometrical  properties  most  employed  in  Plane 
Trigonometry. 

The  angles  at  the  base  of  an  isosceles  triangle  are  equal ; 
and  conversely,  if  two  angles  of  a  triangle  are  equal,  the 
sides  which  subtend  them  are  equal.  (5  and  6.1.) 

The  external  angle  of  a  triangle  is  equal  to  the  two 
opposite  internal  ones.  (32.1.) 

The  three  interior  angles  of  a  triangle  are  equal  to  two 
right  angles  or  180°.  (32.1.) 

Hence,  if  the  sum  of  two  angles  be  subtracted  from  180°, 
the  remainder  will  be  the  third  angle. 

If  one  angle  be  subtracted  from  180°,  the  remainder  is 

O  ' 

the  sum  of  the  other  angles. 

If  one  oblique  angle  of  a  right-angled  triangle  be  sub- 
tracted from  90°,  the  remainder  is  the  other  angle. 

The  sum  of  the  squares  of  the  legs  of  a  right-angled  tri- 
angle is  equal  to  the  square  of  the  hypothenuse.  (47.1.) 

The  angle  at  the  centre  of  a  circle  Fig.  11. 

is  double  the  angle  at  the  circum- 
ference upon  the  same  arc ;  or,  in 
other  words,  the  angle  at  the  cir- 
cumference of  a  circle  is  measured 
by  half  the  arc  intercepted  by  its 
sides.  (20.3.)  Thus,  the  angle  ADB 
is  half  ACB ;  and  is,  therefore,  mea- 
sured by  one-half  of  the  arc  AB. 

The  sides  about  the  equal  angles  of  equiangular  tri- 
angles are  proportionals.  (4.6.) 


SEC.  II.]  DRAFTING   OR   PLATTING.  49 

SECTION  II. 

DRAFTING  OR  PLATTING.* 

114.  DRAFTING  is  making  a  correct  drawing  of  the  parts 
of  an  object.     Platting  is  drawing  the  lines  of  a  tract  of 
land  so  as  correctly  to  represent  its  boundaries,  divisions, 
and  the  various  circumstances  needful  to  be  recorded.     It 
is,  in  fact,  making  a  map  of  the  tract.     It  is  of  great  im- 
portance to  a  surveyor  to  be  able  to  make  a  correct  and 
neat  plat  of  his  surveys.     The  facility  of  doing  so  can  only 
be  acquired  by  practice;  the  student  shouid,  therefore,  be 
required  to  make  a  neat  and  accurate  draft  of  every  pro- 
blem in  Trigonometry  he  is  required  to  solve,  and  of  every 
survey  he  is  required  to  calculate.     It  is  not  sufficient  that 
he  should  draw  a  figure,  as  he  does  in  his  demonstrations  in 
Geometry,  that  will  serve  to  demonstrate  his  principles  or 
afford  him  a  diagram  to  refer  to,  but  he  should  be  obliged 
to  make  all  parts  in  the  exact  proportion  given  by  the  data, 
so  that  he  can,  if  needful,  determine  the  length  of  any  line, 
or  the  magnitude  of  any  angle,  by  measurement. 

115.  Straight  lines.     Straight  lines  are  generally  drawn 
with  a  straight-edged  ruler.     If  a  very  long  straight  line  is 
needed,  a  fine  silk  thread  may  be  stretched  between  the 
points  that  are  to  be  joined,  and  points  pricked  in  the 
paper  at  convenient  distances;  these  may  then  be  joined 
by  a  ruler. 

In  drawing  straight  lines,  care  should  be  taken  to  avoid 
determining  a  long  line  by  producing  a  short  one,  as  any 
variation  from  the  true  direction  will  become  more  mani- 
fest the  farther  the  line  is  produced.  When  it  is  necessary 
to  produce  a  line,  the  ruler  is  fixed  with  most  ease  and  cer- 
tainty by  putting  the  points  of  the  compasses  into  the  line 
to  be  produced,  and  bringing  the  ruler  against  them. 

116.  Parallels.     Parallels  may  be  drawn  as  described  in 

*  Various  hints  in  this  section  have  been  derived  from  Gillespie's  "Laud 
Surveying." 


50 


PLANE  TRIGONOMETRY. 


[CHAP.  Ill 


Arts.  97,  98.     Practically,  however,  it  is  better  to   draw 
them  by  some  instrument  specially  adapted  to  the  purpose. 

The  square  and  ruler  are  very  convenient  instruments 
for  this  purpose.  The  square  consists  of  two  arms,  which 
should  be  made  at  right  angles  to  each  other,  to  facilitate 
the  erection  of  perpendi- 
culars. Let  AB  (Fig.  33)  be 
the  line  to  which  a  parallel 
is  to  be  drawn  through  C. 
Adjust  one  edge  of  the 
square  to  the  line  AB,  and 
bring  a  ruler  firmly  against 
the  other  leg;  move  the 
square  along  the  ruler  un- 
til the  edge  coincides  with 
C :  this  edge  will  then  be 
parallel  to  the  given  line. 

If  a  T  square  be  substituted  for  a  simple  right  angle,  it 
may  be  held  more  firmly  against  the  ruler. 

Instead  of  a  square,  a  right-angled  triangle  is  frequently 
used.  The  legs  should 
be  made  accurately  at 
right  angles,  that  it  may 
be  used  for  drawing  per- 
pendiculars. Let  AB 
(Fig.  34)  be  the  line,  and 
C  the  point  through  which 
it  is  required  to  draw  a 
parallel.  Bring  one  edge  of  the  triangle  accurately  to  the 
line,  and  then  place  a  ruler  against  one  of  the  other  sides. 
Slide  the  triangle  along  the  ruler  until  the  point  C  is  in 
the  side  which  before  coincided  with  the  line :  this  side  is 
then  parallel  to  the  given  line. 

The  parallel  rulers  which  accompany  most  cases  of  in- 
struments are  theoretically  accurate.  They  are,  however, 
generally  made  with  so  little  care  that  they  cannot  be  de- 
pended on  where  correctness  is  required ;  and,  even  if  made 
true,  they  are  liable  to  become  inaccurate  in  consequence 
of  wear  of  the  joints. 


SEC.  II.]  DRAFTING  OR  PLATTING.  51 

117.  Perpendiculars.    Perpendiculars  may  be  drawn  as 
directed,  (Art.  88,  et  seq.)    A  more  ready  means  is  to  place 
one  leg  of  the  square  (Fig.  33)  upon  the  line :  the  other  will 
then  be  perpendicular  to  that  line.     The  triangle  is  another 
very  convenient  instrument  for  this 

purpose.  Let  AB  (Fig.  35)  be  the 
line  to  which  a  perpendicular  is  to 
be  drawn.  Place  the  hypothenuse 
of  the  triangle  coincident  with  AB, 
and  bring  the  ruler  against  one  of 
the  other  sides.  Remove  the  tri- 
angle and  place  it  with  the  third 
side  against  the  ruler,  as  at  D :  then  the  hypothenuse  will  be 
perpendicular  to  AB. 

This  method  requires  the  angle  of  the  triangle  to  be  pre- 
cisely a    right    angle.      To    test  F.   ^ 
whether  it  is   so,  bring   one  leg  B 
against  a  ruler,  as  at  A,  (Fig.  36,) 
and  scribe  the  other  leg.     Reverse 
the  triangle,  and  bring  the  right 
angle  to  the   same  point  A,  and                         A 
again  scribe  the  leg.     If  the  angle  is  a  right  angle,  the  two 
scribes  will  exactly  coincide.    If  they  do  not  coincide,  the 
triangle  requires  rectification. 

118.  Circles  and  Arcs.     These  are  generally  drawn  w^th 
the  compasses,  which  should  have  one  leg  movable,  so  that 
a  pen   or  a  pencil   may  be   inserted  instead  of  a  point. 
When  circles  of  long  radii  are  required,  the  beam  compasses 
should  be  used. 

These  consist  of  a  bar  of  wood  or  metal,  dressed  to  a 
uniform  size,  and  having  two  slides  furnished  with  points. 
These  slides  can  be  adjusted  to  any  part  of  the  beam,  and 
clamped,  by  means  of  screws  adapted  to  the  purpose.  The 
point  connected  with  one  of  the  slides  is  movable,  so  that  a 
pencil  or  drawing  pen  may  be  substituted. 

When  the  beam  compasses  are  not  at  hand,  a  strip  of 
drawing  paper  or  pasteboard  may  be  substituted :  a  pin 
through  one  point  will  serve  as  a  centre;  the  pencil 


52 


PLANE   TRIGONOMETRY. 


[CHAP.  III. 


point   can   be   passed   through,  a  hole    at    the    required 
distance. 

119.  Angles.  Angles  may  be  laid  off  by  a  protractor. 
This  is  usually  a  semicircle  of  metal,  the  arc  of  which  is 
divided  into  degrees.  To  use  it,  place  it  with  the  centre  at 
the  point  at  which  the  angle  is  to  be  made,  and  the  straight 
edge  coincident  with  the  given  line ;  then  with  a  fine  point 
prick  off  the  number  of  degrees  required,  and  join  the  point 
thus  determined  to  the  centre. 

The  figures  on  the  protractor  should  begin  at  each  end 
of  the  arc,  as  represented  in  Fig.  37. 


120.  By  the  Scale  of  Chords.  The  scale  of  chords, 
which  is  engraved  on  the  ivory  scales  contained  in  a  box 
of  instruments,  may  also  be  used  for  making  angles.  For 
this  purpose  take  from  the  scale  the  chord  of  60°  for  a 
radius.  With  the  point  A,  at  which  the  angle  is  to  be  made,  as 
a  centre,  and  that  radius,  describe  an  arc.  Take  off  from 
the  scale  the  chord  of  the  required  number 
of  degrees  and  lay  it  on  the  arc  from  the 
given  line,  join  the  extremity  of  the  arc 
thus  laid  off  to  the  centre,  and  the  thing  is 
done. 

Thus,  if  at  the  point  A  (Fig.  38)  it  were 
required  to  make  an  angle  BAG  of  47°. 


Fig.  38. 


SEC.  II.]  DRAFTING  OR  PLATTING.  53 

"With  the  centre  A  and  radius  equal  to  the  chord  of  60° 
describe  the  arc  BC.  Then,  taking  the  chord  of  47°  from 
the  scale,  lay  it  off  from  B  to  C.  Join  AC,  and  BAG  will 
be  the  required  angle. 

If  an  angle  of  more  than  90°  is  required :  first  lay  off  90°, 
and  from  the  extremity  of  that  arc  lay  off  the  remainder. 

121.  By  the  Table  of  Chords.    The  table  of  chords 
(page  97  of  the  tables)  affords  a  much  more  accurate  means 
of  laying  off  angles. 

Take  for  a  radius  the  distance  10  from  any  scale  of  equal 
parts, — to  be  described  hereafter, — and  describe  the  arc  BC, 
(Fig.  38.)  Then,  finding  the  chord  of  the  required  angle 
by  the  table,  multiply  it  by  10,  and,  taking  the  product 
from  the  same  scale,  lay  it  off  from  B  to  C  as  before.  Join 
AC,  and  the  thing  is  done. 

If  the  angle  is  much  over  60°  it  is  best  to  lay  off  the  60° 
first.  This  is  done  by  using  the  radius  as  a  chord.  The 
remainder  can  then  be  laid  off  from  the  extremity  of  the 
arc  of  60°  thus  determined. 

122.  Distances.    Every  line  on  a  draft  should  be  drawn 
of  such  a  length  as  correctly  to  represent  the  distance  of 
the  points  connected,  in  due  relation  to  the  other  parts  of 
the  drawing.     In  perspective  drawing,  the  parts  are  deline- 
ated so  as  to  present  to  the  eye  the  same  relations  that 
those  of  the  natural  object  do  when  viewed  from  a  particular 
point.     To  produce  this  effect  the  figure  must  be  distorted. 
Right  angles  are  represented  as  right,  obtuse,  or  acute,  ac- 
cording to  the  position  of  the  lines ;  and  the  lengths  of  lines 
are   proportionally  increased  or  diminished   according  to 
their  position.     In  drafting,  on  the  contrary,   every  part 
must  be  represented  as  it  is.     The  angles  should  be  of  the 
same  magnitude  as  they  are  in  reality,  and  the  lines  should 
bear  to  each  other  the  exact  ratio  that  those  which  they 
are  intended  to  represent  do.     The  plat  should,  in  fact,  be 
a  miniature  representation  of  the  figure. 

123.  Drawing  to  a  Scale.     In  order  that  the  due  pro- 


54  PLANE  TRIGONOMETRY.  [CHAP.  III. 

portion  should  exist  in  the  parts  of  the  figure,  every  line 
should  be  made  some  definite  part  of  the  length  of  that 
which  it  is  intended  to  represent.  This  is  called  drawing  to 
a  scale.  The  scale  to  be  used  depends  on  the  size  of  the 
map  or  draft  that  is  required,  and  the  purposes  for  which  it 
is  to  be  used.  Carpenters  often  use  the  scale  of  an  inch  to 
a  foot :  the  lines  will  then  be  the  twelfth  part  of  their  real 
length.  In  plats  of  surveys,  or  maps  of  larger  tracts  of 
country,  a  greater  diminution  is  necessary.  The  scale 
should,  however,  in  all  cases,  be  adapted  to  the  purpose 
intended  and  to  the  number  of  objects  to  be  represented. 
Where  the  purpose  is  merely  to  give  a  correct  representa- 
tion of  the  plat,  without  filling  up  the  details,  the  main 
object  will  be  to  make  the  map  of  a  convenient  size;  but 
where  many  details  are  to  be  represented  the  scale  should 
be  proportionally  larger. 

Thus,  for  example,  in  delineating  a  harbor  where  there 
are  few  obstructions  to  navigation,  a  map  on  a  small  scale 
may  be  drawn ;  but  where  the  rocks  and  shoals  are  nume- 
rous, the  scale  should  be  so  large  that  every  part  may  be 
perfectly  distinct. 

The  scales  on  which  the  drawing  is  made  should  always 
be  mentioned  on  the  map.  They  may  be  expressed  by 
naming  the  lengths  which  are  used  as  equivalents,  thus, — 
"  Scale,  10  feet  to  an  inch,  1  mile  to  an  inch,  3  chains  to 
a  foot;"  or  better  fractionally,  thus,— 1  :  100,  1:250, 
1  :  10,000,  &c. 

124.  Surveys  of  Farms.    Where  the  farm  is  small,  1 

chain*  to  an  inch,  (1 : 792,)  or  2  chains  to  the  inch,  (1 : 1584,) 
may  be  used ;  but  if  the  tract  be  large,  as  this  would  make 
a  plat  of  a  very  inconvenient  size,  a  smaller  scale  must  be 
adopted.  When,  however,  any  calculations  are  to  be  based 
on  measurements  taken  from  the  plat,  a  smaller  scale  than 
3  chains  to  the  inch  (1 :  2376)  should  not  be  employed. 


*  The  surveyor's  chain — commonly  called  Gunter's  Chain — is  4  poles,  or  66 
feet,  in  length,  and  is  divided  into  one  hundred  links,  each  of  which  is  therefore 
.66  feet,  or  7.92  inches  in  length. 


SEC.  II.] 


DRAFTING  OR  PLATTING. 


55 


125.  Scales.  Scales  are  generally  made  of  ivory  or  box- 
wood, having  a  feather-edge,  on  which  the  divisions  are 
marked,  The  distances  can  then  be  laid  off  by  placing  the 
ruler  on  the  line,  and  pricking  the  paper  or  marking  it  with 
a  fine  pointed  pencil ;  or  the  length  of  a  line  may  be  read 
off  without  any  difficulty.  Boxwood  scales,  if  the  wood  is 
clear  from  knots,  are  to  be  preferred  to  ivory.  They  are 
less  liable  to  warp,  and  suffer  less  expansion  and  con- 
traction from  changes  in  the  hygrometric  condition  of  the 
atmosphere. 

Paper  scales  are  often  employed.  These  may  be  pro- 
cured with  divisions  to  suit  almost  any  purpose,  or  the  sur- 
veyor may  make  them  himself.  Take  a  piece  of  drawing- 
paper,  and  cut  a  slip  about  an  inch  in  width ;  draw  a  line 
along  its  middle,  and  divide  it  as  desired,  either  into  inches 
or  tenths  of  a  foot.  The  end  division  should  be  subdivided 
into  ten  parts,  and  perpendiculars  drawn  through  all  the 
divisions,  as  represented  in  the  figure,  (Fig.  39.)  Each  of 
these  parts  may  then  represent  a  chain,  ten  chains,  &c. 

Fig.  39. 


Paper  scales,  being  subject  to  nearly  the  same  expansion 
and  contraction  as  the  paper  on  which  the  map  is  drawn, 
are,  on  this  account,  preferable  to  those  made  of  wood  or 
ivory.  They  cannot,  however,  be  divided  with  the  same 
accuracy. 

126.  The  plane  diagonal  scale  (Fig.  40)  consists  of  eleven 

Fig.  40. 


5                      4 

: 

J                      5 

5                       ] 

A  2 

HP 

p 

46\ 

1  U 

i 

8  B 

s 

^B 

* 

I6 

56  PLANE   TRIGONOMETRY.  [CHAP.  in. 

lines  drawn  parallel  and  equidistant.  These  are  crossed  at 
right  angles  by  lines  1,  2,  3,  drawn  usually  at  intervals  of 
half  an  inch.  The  first  division,  on  the  upper  and  lower 
lines,  is  subdivided  into  ten  equal  parts :  diagonal  lines  are 
then  drawn,  as  in  the  figure,  from  each  division  of  the  top 
to  the  next  on  the  bottom, — the  first,  from  A  to  the  first 
division  on  the  bottom  line;  the  second,  from  the  first  on 
the  top  to  the  second  on  the  bottom ;  and  so  on. 

It  is  evident  that,  whatever  distance  the  primary  division 
from  A  to  1,  or  1  to  2,  &c.  represents,  the  parts  of  the  line 
AB  will  represent  tenth  parts  of  that  distance.  If  then 
it  were  required  to  take  off  the  distance  of  47  feet  on  a 
scale  of  half  an  inch  to  10  feet,  the  compasses  should  be 
extended  from  E  to  F. 

The  diagonal  lines  serve  to  subdivide  each  of  the  smaller 
divisions  into  tenths,  thus: — The  first  diagonal,  extending 
from  A  to  the  first  division  on  the  bottom  line  and  crossing 
ten  equal  spaces,  will  have  advanced  ^  of  one  of  those 
divisions  at  the  first  intermediate  line,  ^  at  the  second,  &  at 
the  third,  and  so  on.  All  the  other  diagonals  will  advance 
in  the  same  manner. 

If  then  the  distance  were  taken  from  the  line  AC  along 
the  horizontal  line  marked  6  to  the  fourth  diagonal,  the 
distance  would  be  .46,  the  division  AB  being  a  unit,  or 
4.6  if  AB  were  10.  To  take  off,  then,  39.8  feet  on  a  scale 
of  half  an  inch  to  10  feet,  the  compasses  should  be  ex- 
tended to  the  points  marked  by  the  arrow  heads  G-  and  H : 
similarly,  46. 7,  on  the  same  scale,  would  extend  from  one 
of  the  arrow  heads  on  the  seventh  line  to  the  other. 

In  using  the  diagonal  scale  the  primary  divisions  should 
always  be  made  to  represent  1, 10, 100,  or  1000.  When  any 
other  scale  is  required, — say  1 :  300, — it  is  better  to  divide 
or  multiply  all  the  distances  and  then  take  off  the  results. 
Thus,  if  83.7  were  required  to  be  taken  off  on  a  scale  of 
J  inch  to  30  feet,  first  divide  83.7  by  3,  giving  27.9,  and 
then  take  off  the  quotient  on  a  scale  of  J  inch  to  10  feet. 
The  other  lines  must  all  be  reduced  in  the  same  proportion. 
The  above  method  requires  less  calculation,  and  involves 


SEC.  II.] 


DRAFTING  OR  PLATTING. 


57 


less  liability  to  error,  than  that  of  determining  the  value  of 
each  division  on  the  reduced  scale. 

127.  Proportional  Scale.  On  most  of  the  rulers  fur- 
nished with  cases  of  instruments  there  is  another  set  of 
scales,  divided  as  below,  (Fig.  41.) 

Fig.  41. 


The  figures  on  the  left  express  the  number  of  divisions  to 
the  inch.  To  lay  off  97  feet  on  a  scale  of  40  feet  to  the 
inch,  the  compasses  would  be  extended  between  the  arrow- 
heads on  the  line  40.  Scales  of  this  kind  are  very  con- 
venient in  altering  the  size  of  a  drawing.  Suppose,  for 
example,  it  is  desired  to  reduce  a  drawing  in  the  ratio  of  5 
to  3 :  the  lengths  of  the  lines  should  be  determined  on  the 
scale  marked  30,  and  the  same  number  of.  divisions  on  the 
scale  50  will  give  a  line  of  the  desired  length. 

128.  Vernier  Scale.  Make  a  scale  (Fig.  42)  with  inches 
divided  into  tenths,  and  mark  the  end  of  the  first  inch  0, 
of  the  second  100,  and  so  on.  From  the  zero  point,  back- 
wards, lay  off  a  space  equal  to  eleven  tenths  of  an  inch, 
and  divide  it  into  ten  equal  parts,  numbering  the  parts 
backwards,  as  represented  in  the  figure.  This  smaller  scale 

Fig.  42. 


1 

00 

2 

00 

~T 

1  M  1 

i  M  1 

1 

1 

1  1 

1     1 

1      1 

1  1 

Vi 

fill 

88 

66 

44         22 

A. 

is  a  vernier.     Now,  since  the  ten  divisions  of  the  vernier 
are  equal  to  eleven  of  the  scale,  each  of  the  vernier  divisions 


58  PLANE   TRIGONOMETRY.  [CHAP.  III. 

is  equal  to  ^  of  ^  =  J^  of  an  inch.  From  the  zero  point, 
therefore,  to  the  second  division  of  the  vernier  is  .22  inch, 
to  the  third  .33,  and  so  on. 

To  measure  any  line  by  the  scale,  take  the  distance  in 
the  compasses,  and  move  them  along  the  scale  until  you 
find  that  they  exactly  extend  from  some  division  on  the 
vernier  to  a  division  on  the  scale.  Add  the  number 
on  the  scale  to  the  number  on  the  vernier  for  the  dis- 
tance required.  Thus,  suppose  the  compasses  extended 
from  66  on  the  vernier  to  110  on  the  scale,  the  length  is 
1T6. 

To  lay  off  a  distance  by  the  scale,  for  example  175,  take 
55  from  175,  and  120  is  left :  extend  the  compass  from  120 
on  the  scale  to  55  on  the  vernier.  To  lay  off  268  =  180  + 
88,  extend  the  compasses  from  180  on  the  scale  to  88  on  the 
vernier,  as  marked  by  the  arrow  heads. 

The  vernier  scale  is  equally  accurate  with  the  diagonal 
scale,  and  much  more  readily  made. 


SECTION  III. 
TABLES  OP  TRIGONOMETRICAL  FUNCTIONS. 

129.  Table  of  Natural  Sines  and  Cosines.  THIS  table 
(page  87  of  the  Tables)  contains  the  sines  and  cosines  to  five 
decimal  places  for  every  minute  of  the  quadrant.  The 
table  is  calculated  to  the  radius  1.  As  the  sine  and  cosine 
are  always  less  than  radius,  the  figures  are  all  decimals.  In 
the  table  the  decimal  point  is  omitted.  If  the  sine  and 
cosine  is  wanted  to  any  other  radius,  the  number  taken 
from  the  table  must  be  multiplied  by  that  radius. 

To  take  out  the  sine  or  cosine  of  an  arc  from  this  table, 
look  for  the  degrees,  if  less  than  45,  at  the  top  of  the  table, 
and  for  the  minutes  at  the  left ;  then,  in  the  column  headed 
properly,  and  opposite  the  minutes,  will  be  the  function 
required.  If  the  degrees  are  45  or  upwards  they  will  be 


SEC.  III.]  TRIGONOMETRICAL  FUNCTIONS.  59 

found  at  the  bottom,  and  the  minutes  at  the  right.  The 
name  of  the  column  is  at  the  bottom. 

Thus,  the  sine  of  32°  17',  found  under  32°  and  opposite 
17',  is  .53411. 

The  cosine  of  53°  24',  found  over  53°  and  opposite  24'  in 
the  right-hand  column,  is  .59622. 

130.  The  table  of  natural  sines  and  cosines  is  of  but  little 
use  in  trigonometrical  calculations,  these  being  generally 
performed  by  logarithms.     It  is  principally  employed  in 
determining  the  latitudes  and  departures  of  lines. 

131.  Table  of  Logarithmic  Sines,  Cosines,  &c.    This 
table  contains  the  logarithms  of  the  sines,  cosines,  tangents, 
and  cotangents,  to  every  minute  of  the  semicircle,  the  radius 
being  10  000  000  000  and  its  logarithm  10.     The  logarithmic 
sine  of  90°,  cosine  of  0°,  tangent  of  45°,  and  cotangent  of 
45°,  is  each  10. 

The  sine,  cosine,  tangent,  and  cotangent,  of  every  arc  being 
equal  to  the  sine,  cosine,  tangent,  and  cotangent,  of  its  supple- 
ment, and  also  to  the  cosine,  sine,  cotangent,  and  tangent,  of  its 
complement,  the  table  is  only  extended  to  forty  five  pages, 
the  degrees  from  0  to  44  inclusive  being  found  at  the  top, 
those  from  45  to  135  at  the  bottom,  and  from  136  to  180  at 
the  top.  The  minutes  are  contained  in  the  two  outer 
columns,  and  agree  with  the  degrees  at  the  top  and  bottom 
on  the  same  side  of  the  page. 

The  columns  headed  Diff.  1"  contain  the  difference  of 
the  function  for  a  change  of  1"  in  the  arc.  These  differ- 
ences are  calculated  by  dividing  the  differences  of  the  suc- 
cessive numbers  in  the  columns  of  the  functions  by  60.  By 
an  inspection  of  these  columns  of  difference  it  will  be  seen 
that,  except  in  the  first  few  pages,  they  change  very  slowly. 
In  these,  in  consequence  of  the  rapid  change  of  the  func- 
tion, the  differences  vary  very  much.  The  difference  set 
down  will  not,  therefore,  be  accurate,  except  for  about  the 
middle  of  the  minute.  The  calculations  for  seconds,  there- 
fore, are  not  in  these  cases  to  be  depended  on.  To  obviate 
this  inconvenience,  and  give  to  the  first  few  pages  a  degree 


60  PLANE    TRIGONOMETRY.  [CHAP.  III. 

of  accuracy  commensurate  with  that  of  the  rest  of  the  table, 
the  sines  and  tangents  are  calculated  to  every  10  seconds, 
and  these  are  the  same  as  the  cosines  and  cotangents  of 
arcs  within  two  degrees  of  90.* 

132.  Use  of  Table.     To  take  out  any  function  from  the 
table,  seek  the  degrees,  if  less  than  45°  or  more  than  135°,  at 
the  top  of  the  page,  and  the  minutes  in  the  column  on  the 
same  side  of  the  page  as  the  degrees.     Then,  in  the  proper 
column,  (the  title  being  at  the  top,)  and  opposite  the  minutes, 
will  be  found  the  value  required. 

If  the  degrees  are  between  45°  and  135°,  seek  them  at 
the  bottom  of  the  page,  the  minutes  being  found,  as  before, 
at  the  same  side  of  the  page  as  the  degrees.  The  titles  of 
the  columns  are  also  at  the  bottom. 

EXAMPLES. 

Ex.  1.  Required  the  sine  of  37°  17'.       Ans.  9.782298. 
Ex.  2.  Required  the  cosine  of  127°  43'.  Ans.  9.786579. 

Ex.  3.  Required  the  cotangent  of  163°  29'. 

Ans.  10.527932. 

Ex.  4.  Required  the  tangent  of  69°  11'. 

Ans.  10.419991. 

133.  If  there  are  seconds  in  the  arc,  take  out  the  function 
for  the  degrees  and  minutes  as  before.     Multiply  the  num- 
ber in  the  difference  column  by  the  number  of  seconds,  and 
add  the  product  to  the  number  first  taken  out,  if  the  func- 
tion  is   increasing,  but  subtract,  if  it  is  decreasing :   the 
result  will  be  the  value  required. 

If  the  arc  is  less  than  90°  the  sine  and  tangent  are  in- 
creasing, and  the  cosine  and  cotangent  are  decreasing ;  but  if 
the  arc  is  greater  than  90°  the  reverse  holds  true. 

*  The  rectangle  of  the  tangent  and  cotangent  of  an  arc  being  equal  to  the 
square  of  radius,  their  logarithms  are  arithmetical  complements  (to  20)  of  each 
other.  Our  column  of  differences  serves  for  both  these  functions.  It  is  placed 
between  them. 


SEC.  III.]  TRIGONOMETRICAL  FUNCTIONS.  61 

Ex.  1.  What  is  the  tangent  of  37°  42'  25"? 

The  tangent  of  37°  42'  is  9.888116 

Diff.  V  4.35 

25 

2175 

870 

Diff.  25"  108.75        +  109 

Tangent  37°  42'  25"  9.888225 

Ex.  2.  What  is  the  cosine  of  129°  17'  53"? 

The  cosine  of  129°  17'  is  9.801511 

Diff.  1"  2.57 

53 
771 
1285 

Diff.  53"  136.21        +136 

Cosine  129°  17'  53"  9.801647 

Ex.  3.  What  is  the  sine  of  63°  19'  23"? 

Ans.  9.951120. 

Ex.  4.  What  is  the  cosine  of  57°  28'  37"? 

Ans.  9.730491. 
Ex.  5.  What  is  the  tangent  of  143°  52'  16"? 

Ans.  9.863314. 
Ex.  6.  What  is  the  sine  of  172°  19'  48"? 

Ans.  9.125375. 

If  the  sine  or  tangent  of  an  arc  less  than  2°  or  more 
than  178°,  or  the  cosine  or  cotangent  of  an  arc  between 
88°  and  92°,  is  required,  it  should  be  taken  from  the  first 
pages  of  the  table.  Take  out  the  function  to  the  ten 
seconds  next  less  than  the  given  arc,  multiply  one  tenth  of 
the  difference  between  the  two  numbers  in  the  table  by  the 
odd  seconds,  and  add  or  subtract  as  before. 

The  cotangent  of  an  arc  less  than  2°  may  be  found  by 
taking  out  the  tangent,  and  subtracting  it  from  20.000000 ; 
so  likewise  the  tangent  of  an  arc  between  178°  and  180° 
is  found  by  taking  the  complement  to  20.000000  of  its 
cotangent. 


62  PLANE   TRIGONOMETRY,  [CHAP.  III. 

Ex.  1.  Required  the  sine  of  1°  27'  36". 

Sine  of  1°  27'  30"  is  8.405687 

&  of  difference  82.6 

6 

Difference  6"  495.6  496 

Sine  of  1°  27'  36"  8.406183 

Ex.  2.  What  is  the  cosine  of  88°  18'  48"? 

Ans.  8.468844. 

Ex.  3.  What  is  the  sine  of  179°  19'  13"? 

Ans.  8.074198. 

134.  To  find  the  Arc  corresponding  to  any  Trigo- 
nometric Function. 

If  degrees  and  minutes  only  be  required,  seek,  in  the  pro- 
per column,  the  number  nearest  that  given ;  and  if  the  title 
is  at  the  top  the  degrees  are  found  at  the  top,  and  the  minutes 
under  the  degrees;  but  if  the  title  is  at  the  bottom  the 
degrees  are  at  the  bottom,  and  the  minutes  on  the  same  side 
as  the  degrees. 

If  seconds  are  desired,  seek  for  the  number  corresponding 
to  the  minute  next  less  than  the  true  arc,  and  take  the 
difference  between  that  number  and  the  given  one :  divide 
said  difference  by  the  number  in  the  difference  column,  for 
the  seconds. 

Ex.  1.  What  is  the  arc  whose  sine  is  9.427586  ? 

9.427586 
Sine  of  15°  31'  is  9.427354 

7.58)232.00(31" 

2274 

4.60 
The  arc  is,  therefore,  15°  31'  31". 


SEC.  III.]  TRIGONOMETRICAL  FUNCTIONS.  63 

Ex.  2.  What  is  the  arc  whose  cotangent  is  10.219684? 

10.219684 
Cotangent  of  31°  5'  is         10.219797 

4.76)  113.00  (23.7" 
952 
1780 
1428 

3.52 
The  arc  is,  therefore,  31°  5'  24". 

Ex.  3.  Required  the  arc  the  cosine  of  which  is  9.764227. 

Ans.  54°  28'  27". 

Ex.  4.  Required  the  arc  the  tangent  of  which  is 
10.876429.  Ans.  82°  25'  44". 

Ex.  5.  What  is  the  arc  the  cotangent  of  which  is  11.562147? 
As  this  corresponds  to  an  arc  less  than  2°,  take  it  from 
20.000000:  the  remainder,  8.437853,  is  the  tangent.     The 
arc  is  found  as  follows : — 

8.437853 

1°  34'  10"  tang.  8.437732 

Diff.  tol"  76.8  )  121.0  (1.6" 

768 
44.20 
The  angle  is,  therefore,  1°  34'  11.6". 

Ex.  6.  What  arc  corresponds  to  the  cotangent  8.164375? 

Ans.  89°  9'  48.6". 

135.  Table  of  Chords.  This  table  contains  the  chords 
of  arcs  to  90°  for  every  5  minutes.  It  is  principally  used 
in  laying  off  angles,  as  explained  in  Art.  120,  and  in  pro- 
tracting surveys  by  the  method  of  Art.  343. 


34  PLANE   TRIGONOMETRY.  [CHAP.  III. 

SECTION  IV. 

ON  THE  NUMERICAL  SOLUTION  OF  TRIANGLES. 

136.  Definition.     THE  solution  of  a  triangle  is  the  deter- 
mination of  the   numerical  value  of  certain  parts  when 
others  are  given.      To   determine  a  triangle,  three  inde- 
pendent parts  must  be  known, — viz. :  either  the  three  sides, 
or  two  sides  and  an  angle,  or  the  angles  and  one  side.     The 
three  angles  are  not  of  themselves  sufficient,  since  they  are 
not  independent, — any  one  of  them  being  equal  to  the  dif- 
ference between  the  sum  of  the  others  and  180°. 

In  the  solution  of  triangles  several  cases  may  be  distin- 
guished ;  these  will  be  treated  of  separately.  These  cases 
are  applicable  to  all  triangles.  But  as  there  are  special 
rules  for  right-angled  triangles,  which  are  simpler  than  the 
more  general  ones,  they  will  first  be  given. 

A.— THE    NUMERICAL   SOLUTION   OF   EIGHT- ANGLED 
TRIANGLES. 

137.  The  following  rules  contain  all  that  is  necessary  for 
solving  the  different  cases  of  right-angled  triangles. 

1.  The  hypothenuse  is  to  either  kg  as  radius  is  to  the  sine  of 
the  opposite  angle. 

2.  The  hypothenuse  is  to  one  leg  as  radius  is  to  the  cosine  of 
the  adjacent  angle. 

3.  One  leg  is  to  the  other  as  radius  is  to  the  tangent  of  the  angle 
adjacent  to  the  former. 

DEMONSTRATION.— Let  ABC  (Fig.  43)  be  a  Fig.  43. 

triangle  right-angled  at  B.  Take  AD  any  ra- 
dius, and  describe  the  arc  DE;  draw  EF  and 
DG  perpendicular  to  AB.  Then  EF  will  be  the 
sine,  AF  the  cosine,  and  DG  the  tangent,  of 
the  angle  A.  Now,  from  simifar  triangles  we 
have — 

1.  AC  :  CB  :  :  AE  :  EF  :  :  r 

2.  AC  :  AB  :  ;  AE  :  AF  :  :  r 

3.  AB  :  BC  :  :  AD  :  DG  :  :  r 


SEC.  IV.]  NUMERICAL  SOLUTION  OF   TRIANGLES.  65 

EXAMPLES. 

Ex.  1.  In  the  triangle  ABC,  right-angled  at  B,  there  are 
given  the  base  AB  =  57.23  chains,  and  the  angle  A  35°  27' 
25",  to  find  the  other  sides. 

Construction. 

Make  AB  (Fig.  44)=  57.23,  taken  *fe-  44. 

from  a  scale  of  equal  parts.  At  the 
point  A  make  the  angle  BAG  = 
35°  27'.  Erect  the  perpendicular 
BC,  meeting  AC  in  C,  and  ABC 
is  the  triangle  required. 


Calculation. 

Rule  3.     r  :  tan.  A  : :  AB  :  BC. 
Eule  2.     cos.  A  :  r  : :  AB  :  AC. 

For  facility  of  calculation,  the  proportions  are  generally 
written  vertically,  as  below. 

As  rad.  log.  10.000000 

:    tan.  A  35°  27'  25"  9.852577 

: :  AB  57.23  ch.  1.757624 

:    BC  40.76  1.610201 

As  cos.  A          35°  27'  25"  Ar.  Co.          0.089081 
:    rad.  10.000000 

::  AB  57.23  1.757624 

:    AC  70.26  1.846705 

Ex.  2.  Given  AB  =  47.50  chains,  and  AC  =  63.90  chains, 
to  find  the  angles  and  side  BC. 

EULE  2. 

As  AC        63.90     Ar.  Co.  8.194499 
:  AB        47.50  1.676694 

::  rad.  10.000000 

:  cos.  A      41°  58'  57"       9.871193 
90 

C       48°  V  3" 
5 


66  PLANE   TRIGONOMETRY.  [CHAP.  III. 

RULE  1. 

As  rad.  10.000000 

:  sin.  A  41°  58'  57"  9.825363 

::  AC  63.90  1.805501 

:  CB  42.74  "l.630864 

Ex.  3.  Given  the  two  legs  AB  =  59.47  yards,  and  BO  = 
48.52  yards,  to  find  the  hypothenuse  and  the  angles. 
Ans.  A  39°  12'  36",  C  50°  47'  24",  and  AC  76.75  yds. 

Ex.  4.  Given  the  hypothenuse  AC  =  97.23  chains,  the 
perpendicular  BC  =  75.87  chains,  to  find  the  rest. 

Ans.  A  51°  17'  22",  C  38°  42'  38",  AB  60.81  ch. 

Ex.  5.  Given  the  angle  A  =  42°  19'  24",  and  the  perpen- 
dicular BC  =  25.54  chains,  to  find  the  other  sides. 

Ans.  AC  37.932  ch.,  AB  28.045  ch. 

Ex.  6.  Given  the  angle  C  =  72°  42'  9",  and  the  hypo- 
thenuse AC  =  495  chains,  to  find  the  other  sides. 

Ans.  AB  472.612  ch.,  BC  147.18  ch. 

Ex.  7.  In  the  right-angled  triangle  ABC  we  have  the 
base  AB  =  63.2  perches,  and  the  angle  A  42°  8'  45",  to 
find  the  hypothenuse  and  the  perpendicular. 

Ans.  BC  57.20  p.,  AC  85.24  p. 

138,  "When  two  sides  are  given,  the  third  may  be  found 
by  (47.1);  thus, 

1.  Given  the  hypothenuse  and  one  leg,  to  find  the  other. 

Rule.  From  the  square  of  the  hypothenuse  subtract  the  square 
of  the  given  leg:  the  square  root  of  the  remainder  will  be  the 
other  leg  ;  or, 

Multiply  the  sum  of  the  hypothenuse  and  given  leg  by  their 
difference :  the  square  root  of  this  product  will  be  the  other  leg. 

This  is  evident  from  (47.1)  and  (cor.  5.2.) 

2.  Given  the  two  legs,  tojind  the  hypothenuse. 

Rule.  Add  the  squares  of  the  two  legs,  and  extract  the  square 
root  of  the  sum:  the  result  will  be  the  hypothenuse. 


SEC.  IV.]  NUMERICAL  SOLUTION  OF   TRIANGLES.  67 

EXAMPLES. 

Ex.  1.  Given  the  hypothenuse  AC  =  45  perches,  and  the 
leg  BC  =  29  perches,  to  find  the  other  leg. 


Kule  1.  AB  =  >/AC2  -  BC2  =  N/2025  -  841  =  ^1184 
34.41. 


or,  AB  =  x/(AC  +  BC).(AC  -  BC)  =  ^74  x  16  = 

^1184=  34.41. 

Ex.  2.  The  two  legs  AB  and  AC  are  6  and  8  respectively : 
what  is  the  hypothenuse  ?  Ans.  10. 

Ex.  3.  The  hypothenuse  AC  is  47.92  perches,  and  the 
leg  AB  is  29.45  perches :  required  the  length  of  BC. 

Ans.  37.8  perches. 

Ex.  4.  The  hypothenuse  of  a  right-angled  triangle  is 
49.27  yards,  and  the  base  37.42  yards :  required  the  perpen- 
dicular. Ans.  32.05. 


B.— THE  NUMERICAL  SOLUTION  OF  OBLIQUE-ANGLED 
TRIANGLES. 

CASE  1. 

139.   The  angles  and  one  side,  or  two  sides  and  an  angle  oppo- 
site to  one  of  them,  being  given,  to  find  the  rest. 

EULE. 

1.  As  the  sine  of  the  angle  opposite  the  given  side  is  to  the 
sine  of  the  angle  opposite  the  required  side,  so  is  the  given  side 
to  the  required  side. 

2.  As  the  side  opposite  the  given  angle  is  to  the  other  given 
side,  so  is  the  sine  of  the  angle  opposite  to  the  former  to  the  sine 
of  the  angle  opposite  the  latter. 

DEMONSTRATION. — Both  the  above  rules  are  combined  in  the  general  propo- 
sition.    The  sides  are  to  one  another  as  the  sines  of  their  opposite  angles. 


Let  ABC  (Fig.  45)  be  any  triangle.     From  C  let  fall  / 

CD  perpendicular  to  AB.     Then  (Art.  137)  AC  :  CD  : :  r  / 

:  sin.  A,  and  CD  :  CB  : :  sin.  B  :  r.     Whence  (23.5)  AC  :         / 
CB  : :  sin.  B  :  sin.  A.  / 


68 


PLANE  TRIGONOMETRY. 


[CHAP.  III. 


EXAMPLES. 

Ex.  1.  In  the  triangle  ABC  are  given  AB  =  123.5,  the 
angle  B  =  39°  47'  20",  and  C  =  74°  52'  10":  required  the 
rest. 

Construction. 

The  angle  A  =  180  -  (B  -f-  C)  =  180°  -  114°  39'  30"  = 
65°  20'  30". 

Draw  AB  (Fig.  45)  =  123.5.  At  the  points  A  and  B 
draw  AC,  BC,  making  the  angles  BAC  and  ABC  equal, 
respectively,  to  65°  20'  30"  and  39°  47'  20"  ;  then  will  ABC 
be  the  triangle  required. 


As  sin.  C 
:    sin.  B 

:  :  AB 
:   AC 

As  sin.  C 
:    sin.  A 
::  AB 

:    BC 


Calculation. 
74°  52'  10" 
39°  47'  20" 
123.5 

81.87 

65°  20'  30" 
116.27 


A.  C.  0.015322 
9.806154 
2.091667 
1.913143 

A.  C.  0.015322 
9.958474 
2.091667 
2.065463 


Ex.  2.  Given  the  side  AB  =  327,  the  side  BC 
the  angle  A  =  32°  27',  to  determine  the  rest. 


238,  and 


Construction. 

Make  AB  (Fig.  46)  =  327;  and 
at  the  point  A  draw  AC  making 
the  angle  A  =  32°  47'.  With  the 
centre  B  and  radius  =  238  describe 
an  arc  cutting  AC  in  C ;  then  will 
ABC  be  the  triangle  required. 

Calculation.    EULE  2. 


Kg' 


As  BC 

:    AB 
:  :  sin.  A 

:    sin.  C 
or 


238 
327 

32°  47' 
48°    4' 


6" 


A.  C.  7.623423 
2.514548 
9.733569 

9.871540 


131°  55'  54" 


SEC.  IV.]          NUMERICAL  SOLUTION  OF  TRIANGLES.  69 

C  acute. 

As  sin.  C                48°  4'    6"  A.  C.  0.128460 

:    sin.  B                99°  8'  54"  9.994441 

: :  AB                  327  2.514548 

:   AC                 433.9T  2.637459 

C  obtuse. 

As  sin.  C  131°  55'  54"  A.  C.  0.128460 

:  sin.  B  15°  IT'  6"  9.4209T9 

::  AB  2.514548 

:  AC  115.87  2.063987 

NOTE. — It  will  be  seen  that  in  the  above  example  the  result  is  uncertain. 
The  sine  of  an  angle  being  equal  to  the  sine  of  its  supplement,  it  is  impossible, 
from  the  sine  alone,  to  determine  whether  the  angle  should  be  taken  acute  or 
obtuse.  By  reference  to  the  construction,  (Fig.  46,)  we  see  that  whenever  the 
side  opposite  the  given  angle  is  less  than  the  other  given  side,  and  greater  than 
the  perpendicular  BD,  the  triangle  will  admit  of  two  forms:  ABC,  in  which 
the  angle  opposite  to  the  side  AB  is  acute,  and  ABC'',  in  which  it  is  obtuse. 
If  BC  were  greater  than  BA,  the  point  G/  would  fall  on  the  other  side  of  A, 
and  be  excluded  by  the  conditions.  If  it  were  less  than  BD,  the  circle  would 
not  meet  AC,  and  the  question  would  be  impossible. 

Ex.  3.  Given  the  side  AB  37.25  chains,  the  side  AC  = 
42.59  chains,  and  the  angle  C  57°  29'  15",  to  determine 
the  rest. 

Ans.  BC  32.774  chains,  A  =  47°  53'  52",  and  B  =  74° 
36'  53". 

Ex.  4.  Given  the  angle  A  29°  47'  29",  the  angle  B  =  24° 
15'  17",  and  the  side  AB  325  yards,  to  find  the  other  sides. 
Ans.  AC  =  164.93,  BC  =  199.48. 

Ex.  5.  The  side  AB  of  an  obtuse-angled  triangle  is 
127.54  yards,  the  side  AC  106.49  yards,  and  the  angle 
B  52°  27'  18",  to  determine  the  remaining  angles  and  the 
side  BC. 

Ans.  C  =  108°  16'  3",  A  =  19°  16'  39",  BC  =  44.34. 

Ex.  6.  Given  AB  =  527.63  yards,  AC  =  398.47  yards, 
and  the  angle  B  43°  29'  11",  to  determine  the  rest. 

Ans.  C  =    65°  40'  44",  A  =  70°  50'    5",  BC  =  546.93; 
or,    C  =  114°  19'  16",  A  =  22°  11'  33",  BC  =  218.71. 


70  PLANE   TRIGONOMETRY.  [CHAP.  Ill 

CASE  2. 

140.  Two  sides  and  the  included  angle  being  given,  to  determine 
the  rest. 

EULE   1. 

Subtract  the  given  angle  from  180°  :  the  remainder  will  be  the 
sum  of  the  remaining  angles.  Then, 

As  the  sum  of  the  given  sides  is  to  their  difference,  so  is  the 
tangent  of  half  the  sum  of  the  remaining  angles  to  the  tangent 
of  half  their  difference. 

This  half  difference  added  to  the  half  sum  will  give  the  angle 
opposite  the  greater  side,  and  subtracted  from  the  half  sum  will 
give  the  angle  opposite  the  less  side. 

Then  having  the  angles,  the  remaining  side  may  be  found  by 
Case  1. 

DEMONSTRATION. — The  second  paragraph  of  this  rule  may  be  enunciated  in 
general  terms ;  thus, 

As  the  sum  of  two  sides  of  a  plane  triangl&is  to  Fig.  47. 

their  difference,  so  is  the  tangent  of  half  the  sum  of 
the  angles  opposite  those  sides  to  the  tangent  of  half 
the  difference  of  those  angles. 

Let  ABC  (Fig.  47)  be  the  triangle  of  which  the 
side  AC  is  greater  than  AB.  With  the  centre  A 
and  radius  AC  describe  a  circle  cutting  AB  pro- 
duced in  E  and  F.  Join  EC  and  CF,  and  draw 
FG  parallel  to  BC.  Then,  because  ABC  and  AFC 
have  the  common  angle  A,  AFC  -f-  ACF  =  ABC 
-f  ACB.  Whence  AFC  =  &  (ABC  +  ACB)  ;  and, 

since  the  half  sum  of  two  quantities  taken  from  the  greater  leaves  their  half 
difference,  CFG  =  EFG  —  EFC  ==  ABC  —  EFC  =  £  (ABC  —  ACB). 

Now,  since  the  angle  ECF  is  an  angle  in  a  semicircle,  it  is  a  right  angle. 
Therefore,  if  with  the  centre  F  and  radius  FC  an  arc  be  described,  EC  and 
CG  will  be  the  tangents  of  EFC  and  CFG,  or  of  the  half  sum  and  half  dif- 
ference of  ABC  and  ACB.     But  (2.6)  EB  :  BF  : :  EC  :  CG. 
Whence  AC  -f  AB  :  AC  —  AB  : :  tan.  £  (ABC  -f  ACB)  :  tan.  £  (ABC  —  ACB). 

EXAMPLES. 

Ex.  1.  Given  AJB  =  527  yards,  AC  =  493  yards,  and  the 
angle  A  =37°  49'. 

Here        C  +  B  =  180°  -  37°  49'  =  142°  11',  and 


SEC.  IV.]  NUMERICAL  SOLUTION  OP  TRIANGLES.  71 


As  AB  +  AC 

1020 

A.C.  6.991400 

:  AB-AC 

34 

1.531479 

C  +  B 

:  :  tan.  —  -  — 

71°    5'  30" 

10.465290 

2 

C-B 

:  tan.  —  -  — 

5°  33'  29" 

8.988169 

2 

C 

76°  38'  59" 

B 

65°  32'    1" 

As  sin.  C 

76°  38'  59" 

A.C.  0.011897 

:  sin.  A 

37°  49' 

9.787557 

::AB 

527 

2.721811 

:  BC 

332.10 

2.521265 

Ex.  2.  In  the  triangle  ABC  are  given  AB  =  1025.57  yaids, 
BC  =  849.53  yards,  and  the  angle  B  =  65°  43'  20",  to  find 
the  rest. 

Ans.  A  =  48°  52'  10",  C  =  65°  24'  30",  AC  =  1028.13. 

Ex.  3.  Two  sides  of  a  triangle  are  155.96  feet  and 
217.43  feet,  and  their  included  angle  49°  19',  to  find  the 
rest. 

Ans.  Angles,  85°  4'  12",  45°  36'  48",  side,  165.49. 

RULE  2. 

141.  As  the  less  of  the  two  given  sides  is  to  the  greater,  so  is 
radius  to  the  tangent  of  an  angle;  and  as  radius  is  to  the  tangent 
of  the  excess  of  this  angle  above  45°,  so  is  the  tangent  of  the 
half  sum  of  the  opposite  angles  to  the  tangent  of  their  half 
difference. 

Having  found  the  half  difference,  proceed  as  in  Rule  1. 

NOTE. — This  rule  is  rather  shorter  than  the  last,  where  the  two  sides  have 
been  found  in  a  preceding  calculation,  and  thus  their  logarithms  are 
known. 


r2  PLANE  TRIGONOMETRY.  [CHAP.  III. 

DEMONSTRATION. — Let    ABC    (Fig.  48)    be    any  Fig.  48. 

plane  triangle.  Draw  BD  perpendicular  to  AB,  the 
greater,  and  equal  to  BC,  the  less  side.  Make  BE  = 
BD,  and  join  ED.  Then,  since  BE  =  BD,  the  angle 
BED  =B  BDE ;  and  since  EBD  is  a  right  angle,  BDE 
ss  45°.  But  BED  +  BDE  =  2  BDE  =  BAD  + 
BDA,  and  BDE  =  }  (BDA  +  BAD).  But  the  half 
sum  of  any  two  quantities  being  taken  from  the 
greater  will  leave  the  half  difference:  therefore 
ADE  is  the  half  difference  of  BDA  and  BAD. 

Now,  (Rule  3,  Art.  137,)  BD  or  BC  :  BA  :  :  rad.  :  tan.  ADB ; 

and  (demonstration  to  last  rule)  AB  -j-  BD  :  AB  —  BD  :  :  tan.  \  (BDA  -f- 
BAD)  :  tan.  \  (BDA— BAD)  :  :  tan.  BDE  :  tan.  ADE;  but  BDE  being  equal 
to  45°,  its  tangent  =  rad. 

And  ADE  SB  (ADB  —  45°)  .  •.  AB  +  BD  :  AB  —  BD  : :  r  :  tan.  (ADB  —  45°) ; 
but  AB  +  BC  :  AB  —  BC  :  :  tan.  £  (ACB  +  BAG)  :  tan.  £  (ACB  —  BAC) ; 
whence  r  :  tan.  (ADB  —  45°)  :  :  tan.  J  (ACB  -f  BAC)  :  tan.  }  (ACB  —  BAC). 

EXAMPLES. 

Ex.  1.  In  the  course  of  a  calculation  I  have  found  the 
logarithm  of  AB  =  2.596387,  that  of  BC  =  2.846392:  now, 
the  angle  B  being  55°  49',  required  the  side  AC. 

Calculation. 

AsAB  A.  C.  7.403613 

:  BC  2.846392 

: :  Kad.  10.000000 

:  tan.  x         60°  38'  58"  10.250005 


As  rad.  A.  C.  0.000000 

:  tan.  (x  -  45)  15°  38'  58"  9.447368 

::tan.  J(A+C)  62°  5' 30"  10.276004 

:  tan.  J  (A  -  C)  27°  52'  28"  9.723372 

A  89°  57'  58" 
Then, 

As  sin.  A  89°  57'  58"  A.  C.  0.000000 

:  sin.  B  55°  49'  9.917634 

::BC  2.846392 

:  AC  580.8  2.764026 


SEC.  IV.]  NUMERICAL  SOLUTION  OF  TRIANGLES.  73 

Ex.  2.  Given  the  logarithms  of  BC  and  AC  3.964217 
and  3.729415  respectively,  and  the  angle  C  =  63°  17'  24",  to 
find  AB.  Ans.  8317. 

Ex.  3.  Given  the  logarithms  of  AB  and  BC  1.963425  and 
2.416347,  and  the  angle  B  =  129°  42',  to  find  AC. 

Ans.  327.27. 

CASE  3. 

142.  Given  the  three  sides,  to  find  the  angles. 

EULB  1. 

Call  the  longest  side  the  base,  and  on  it  let  fall  a  perpendicular 
from  the  opposite  angle. 

Then,  as  the  base  is  to  the  sum  of  the  other  sides,  so  is  the 
difference  of  those  sides  to  the  difference  of  the  segments  of  the 
base. 

Half  this  difference  added  to  half  the  base  will  give  the  greater 
segment,  and  subtracted  will  give  the  less  segment. 

Having  the  segments  of  the  base,  and  the  adjacent  sides, 
the  angles  may  be  found  by  Kule  2,  Art.  137. 

DEMONSTRATION.—  Let  ABC  (Fig.  49)  be  the  tri-  Fig.  49. 

angle,  AB  being  the  longest  side :  with  the  centre 
C  and  a  radius  CB,  the  less  of  the  other  sides, 
describe  a  circle,  cutting  AB  in  E  and  AC  in  F 
and  G.  Draw  CD  perpendicular  to  AB.  Then 
(3.3)  DE  =  DB;  therefore  AE  is  the  difference 
of  the  segments  of  the  base. 


Also,  AG  =  AC  +  CB ;  and  AF  =  AC  —  CB. 

Now,  (36.3.  cor.,)  AB  .  AE  =  AG .  AF; 

whence  (16.6)  AB  :  AG  :  :  AF  :  AE, 

or  AB  :  AC  -f  CB  : :  AC  —  CB  :  AD  — DB. 

EXAMPLES. 

Ex.  1.  Given  the  three  sides  of  a  triangle, — viz. :  AB  = 
467,  AC  =  413,  and  BC  =  394,  to  find  the  angles. 


74  PLANE  TRIGONOMETRY.  [CHAP.  III. 

As   AB  467                  Ar.  Co.  7.330683 

:  AC  +  BC  807                                2.906874 

: :  AC  -  BC  19                               1.278754 

:  AD-DB  32.833                         1.516311 

i  (AD-DB)  16.4165 

JAB  233.5 

AD  249.9165 

BD  217.0835 

As  AC  413  Ar.  Co.  7.384050 

:  AD  249.9165  2.397794 

::r  10.000000 

:  cos.  A  52°  45'  44"  9.781844 

As  BC  394  Ar.  Co.  7.404504 

:  BD  217.0835  2.336627 

::r  10.000000 

:    cos.  B  56°  33'  58"  9.741131 

Whence  C  =  180  -  (A+  B)  =  70°  40'  18". 

Ex.  2.  Given  the  three  sides  of  a  triangle,  BC  167,  AB 
214,  and  AC  195  yards,  respectively,  to  find  the  angles. 
Ans.  A  =  47°  55'  13",  B  =  60°  4'  19,  C  =  72°  0'  28". 

Ex.  3.  Given  AB  =  51.67,  AC  =  43.95,  and  BC  =  27.16, 

to  find  the  angles. 

Ans.  A  =  31°  42'  42",  B  =  58°  16'  34",  C  =  90°  0'  44". 

KULE  2. 

143.  As  the  rectangle  of  two  sides  is  to  the  rectangle  of  the 
half  sum  of  the  three  sides  and  the  excess  thereof  above  the  third 
side,  so  is  the  square  of  radius  to  the  square  of  the  cosine  of  half 
the  angle  contained  by  the  first  mentioned  sides. 


SEC.  IV.]  NUMERICAL  SOLUTION  OF  TRIANGLES.  75 

DEMONSTRATION.  —  Let  ABC  (Fig.  50)  Fig.  50. 

be  a  triangle,  of  which  AB  is  greater 
than  AC.  Make  AD  =  AC.  Join  DC, 
and  bisect  it  by  AEF.  Draw  EH  paral- 
lel and  equal  to  CB.  Join  HB,  and  pro- 
duce it  to  meet  AEF  in  F.  Then,  since  j 
EH  is  equal  and  parallel  to  CB,  BH  is 
equal  and  parallel  to  CE,  (33.1.) 
Therefore  F  is  a  right  angle.  Again: 
since  BH  is  equal  to  ED,  and  the  angle 
EGD  =  BGH  and  EDG  =  GBH,  (26.1,)  DG  =  GB  and  EG  =  GH.  On  EH 
describe  a  circle,  and  it  will  pass  through  F. 

Now,  2  AK  =  2  AG-f-  2  GK  =  AC  -f  AD-f  2  DG  -f  2  GK  =  AC  -f  AB-f  BC  ; 
or  AK  =  £  (AC  +  AB  -f  BC), 

and  AI  =  AK—  KI  =  J  (AC  +  AB  +  BC)  —  BC. 

But,  (Rule  2,  Art.  137,)  As     AD  :  AE  :  :  r  :  cos.  DAE  (cos.  £  BAC), 
and  AB  :  AF  :  :  r  :  cos.   %  BAC  ; 

whence  (23.6)          AB  .  AD  :  AE  .  AF  :  :  r1  :  cos.a  \  BAC. 

But  (36.3,  Cor.)  AE  .  AF  =  AK  .  AI  =  £  (AC  +  AB  +  BC)  .  $(A.C  +  AB  -f 
BC)  —  BC; 

whence  AB  .  AC  :  £  (AC  +  AB  +  BC)  •  (£  (AC  -f  AB  +  BC)—  BC)  :  :  r1  :  cos.a  \  BAC. 

EXAMPLES. 

Ex.  1.  Given  AB  =  467,  AC  =  413,  and  BC  =  394,  to  find 
the  angle  C. 

Here,  put  s  —  half  sum  of  the  sides  :  we  have  s  =  637  and 
s  —  AB  =  170;  whence 


AC  BcIA°        41B 

A.C.    7.384050 

J  \  BC        394 

A.C.    7.404504 

,      ATf,  }s            637 
;  s.(5—  AB)  <         *  -r>  -.^r/% 
'  \s-AB  170 

2.804139 
2.230449 

::R2 

20.000000 

:  cos.2J  BCA 

2)19.823142 

J  BCA  =        35°  20;    9'' 

9.911571 

BCA  =        70°  40'  18". 

In  the  above  calculation  the  R*  and  its  logarithm  might  have  been  omitted, 
since  we  have  to  deduct  20  in  consequence  of  having  taken  two  arithmetical 
complements.  The  sum  of  the  logarithms  is  divided  by  2,  to  extract  the  square 
root,  (Art.  16.) 


76  PLANE  TRIGONOMETRY.  [CHAP.  III. 

The  rule  may  be  expressed  thus : — 

Add  together  the  arithmetical  complements  of  the  logarithms 
of  the  two  sides  containing  the  required  angle,  the  logarithm 
of  the  half  sum  of  the  three  sides,  and  the  logarithm  of  the  excess 
of  the  half  sum  above  the  side  opposite  to  the  required  angle :  the 
half  sum  of  these  four  logarithms  will  be  the  logarithmic  cosine 
of  half  that  angle. 

Ex.  2.  Given  AE  =  167,  AC  =  214,  and  EC  =  195,  to  find 
the  angles. 

Ans.  A  =  60°  4'  22",  B  =  72°  0'  28",  C  =  47°  55'  16". 

Ex.  3.  Given  AB  =  51.67,  AC  =  43.95,  and  BC  =  27.16,  to 
find  the  angles. 

Ans.  A  =  31°  42' 40",  B  =  58°  16'  28",  C  =  90°  0'  52". 


SECTION  T. 
INSTRUMENTS  AND  FIELD  OPERATIONS, 

144,  The  Chain.  GUNTEE'S  CHAIN  is  the  instrument 
most  commonly  employed  for  measuring  distances  on  the 
ground.  For  surveying  purposes,  it  is  made  66  feet  or  4 
perches  long,  and  is  formed  of  one  hundred  links,  each  of 
which  is  therefore  .66  feet  or  7.92  inches  long.  The  links 
are  generally  connected  by  two  or  three  elliptic  rings,  to 
make  the  chain  more  flexible.  A  swivel  link  should  be 
inserted  in  the  middle,  that  the  chain  may  turn  without 
twisting.  In  order  to  facilitate  the  counting  of  the  links, 
every  tenth  link  is  marked  by  a  piece  of  brass,  having  one, 
two,  three,  or  four  points,  according  to  the  number  of  tens, 
reckoned  from  the  nearest  end  of  the  chain.  Sometimes 
the  number  of  links  is  stamped  on  the  brass.  The  middle 
link  is  also  indicated  by  a  round  piece  of  brass. 

The  advantage  of  having  a  chain  of  this  particular  length 
is,  that  ten  square  chains  make  an  acre.  The  calculations 


SEC.V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  77 

are  therefore  readily  reduced  to  acres  by  simply  shifting 
the  decimal  point.  There  being  one  hundred  links  to  the 
chain,  all  measures  are  expressed  decimally,  which  renders 
the  calculations  much  more  convenient.  Eighty  chains 
make  one  mile. 

In  railroad  surveying,  a  chain  of  one  hundred  feet  long 
is  preferred,  the  dimensions  being  thus  at  once  given  in 
feet. 

When  the  measurements  are  required  to  be  made  with 
great  accuracy,  rods  of  wood  or  metal,  which  have  been 
made  of  precisely  the  length  intended,  are  used.  In  the 
surveys  of  the  American  Coast  Survey,  the  unit  of  length 
employed  is  the  French  metre,  equal  to  the  10000000th  part 
of  the  quadrant  of  the  meridian.  The  metre  is  39.37079 
inches  =  3.280899  feet  =  1.093633  yards  long. 

It  were  much  to  be  desired  that  the  metre,  or  some  other 
unit  founded  on  the  magnitude  of  the  earth,  or  on  some 
other  natural  length,  such  as  that  of  a  pendulum  beating 
seconds  at  a  given  latitude,  were  universally  adopted  as  the 
unit.  The  metre  will  probably  gradually  come  into  general 
use. 

To  reduce  chains  and  links  to  feet,  express  the  links 
decimally  and  multiply  by  66.  Thus,  7  chains  57  links  = 
7.57  chains  are  equal  to  7.57  X  66  =  499.62  feet  =  499  feet 
7.4  inches. 

To  reduce  feet  and  inches  to  chains,  divide  by  66,  or  by  6 
and  11.  The  inches  must  first  be  reduced  to  a  decimal  of  a 

rr»o  f*t7 

foot,  'thus,  563  feet  8  inches  =  563. 67  feet  =  -  —  ch.  = 
8.54  chains. 

Instead  of  a  chain  of  66  feet,  one  of  33  feet,  divided  into 
fifty  links,  is  sometimes  used.  This  is  really  a  half  chain, 
and  should  be  so  recorded  in  the  notes.  The  half  chain  is 
more  convenient  when  the  ground  to  be  measured  is 
uneven. 

145.  The  chain  is  liable  to  become  incorrect  by  use;  its 
connecting  rings  may  be  pulled  open,  and  thus  the  chain 
become  too  long,  or  its  links  may  be  bent,  which  will 


78  PLANE   TRIGONOMETRY.  [CHAP.  III. 

shorten  the  chain.  Every  surveyor  should,  therefore,  have 
a  carefully  measured  standard  with  which  to  compare  his 
chain  frequently.  According  to  the  laws  of  Pennsylvania, 
such  a  standard  is  directed  to  be  marked  in  every  county 
town,  and  all  surveyors  are  required  to  compare  their  chain 
therewith  every  year. 

If  the  chain  is  too  long,  it  may  be  shortened  by  tighten- 
ing the  rings ;  if  it  is  too  short,  which  it  can  only  become 
by  some  of  the  links  having  been  bent  or  some  rings 
tightened  too  much,  these  should  be  rectified. 

It  has  been  found  that  a  distance  measured  by  a  perfectly 
accurate  chain  is  very  generally  recorded  too  long;  if  then 
the  chain  is  found  slightly  too  long,  say  from  one  fourth 
to  one  third  of  an  inch,  it  need  not  be  altered,  a  distance 
measured  with  such  a  chain  being  more  accurately  recorded 
than  if  the  chain  were  correct. 

In  using  the  chain,  care  should  be  taken  to  stretch  it 
always  with  the  same  force,  or  the  different  parts  of  the  line 
will  not  be  correctly  recorded.  Like  all  other  instruments, 
it  should  be  carefully  handled,  as  it  is  liable  to  injury. 

146.  The  Fins.    In  using  the  chain,  ten  pins  are  necessary 
to  set  in  the  ground  to  mark  the  end  of  each  chain  measured. 
These  are  usually  made  of  iron,  and  are  about  a  foot  or  fif- 
teen inches  long,  the  upper  end  being  formed  into  a  ring, 
and.  the  lower  sharpened  that  they  may  be  readily  thrust 
into  the  ground.     Pieces  of  red  and  white  cloth  should  be 
tied  to  the  ring,  to  distinguish  them  when  measuring  through 
grass  or  among  dead  leaves. 

147.  Chaining.     This  operation  requires  two  persons. 
The  leader  starts  with  the  ten  pins  in  his  left  hand  and 
the  end  of  the  chain  in  his  right;   the  follower,  remain- 
ing at  the  starting  point  and  looking  at  the  staff  set  up 
to  mark  the  other  end  of  the  line,  directs  the  leader  to 
extend  the  chain  precisely  in  the   proper  direction.      The 
leader  then  sticks  one  pin  perpendicularly  into  the  ground 
at  the  end  of  the  chain.     They  then  go  on  until  the  follower 
comes  to  this  pin,  when  he  again  puts  the  leader  in  line, 


SEC.  V.]  INSTRUMENTS  AND   FIELD   OPERATIONS.  79 

who  places  a  second  pin.  The  follower  then  takes  up  the 
first  pin,  and  the  same  operation  is  repeated  until  the  leader 
has  expended  all  his  pins.  "When  he  has  stuck  his  last 
pin,  he  calls  to  the  follower,  who  comes  forward,  bringing 
the  pins  with  him.  The  distance  measured — viz. :  ten 
chains — is  then  noted.  The  leader,  taking  all  the  pins,  again 
starts,  and  the  operation  is  repeated  as  before.  When  the 
leader  has  arrived  at  the  end  of  the  line,  the  number  of  pins 
in  possession  of  the  follower  shows  the  number  of  chains 
since  the  last  "out,"  and  the  number  of  links  from  the  last 
pin  to  the  end  of  the  line,  the  number  of  odd  links.  Thus, 
supposing  there  were  two  "outs,"  and  the  follower  has  six 
pins,  the  end  of  the  line  being  27  links  from  the  last  pin, 
the  length  would  be  26.27  chains. 

Some  surveyors  prefer  eleven  pins.  One  pin  is  then 
stuck  at  the  beginning  of  the  line,  and  at  every  "out"  a 
pin  is  left  in  the  ground  by  the  leader. 

If  the  chain-men  are  both  equally  careful,  they  may 
change  duties  from  time  to  time.  If  otherwise,  the  more 
intelligent  and  careful  man  should  act  as  follower,  that 
being  much  the  more  responsible  position. 

148,  Recording  the  "  Outs."  As  every  "  out"  indicates 
ten  chains, — or  five  chains,  if  a  two-pole  chain  is  used, — it  is 
of  great  importance  to  have  them  carefully  kept.  Various 
contrivances  have  been  suggested  for  that  purpose.  Some 
chain -men  carry  a  string,  in  which  they  tie  a  knot  for  every 
out ;  others  place  in  one  pocket  a  number  of  pebbles,  and 
shift  one  to  another  pocket  at  each  out.  Either  of  these 
methods  is  sufficient  if  faithfully  followed  out.  One  rule, 
however,  should  be  faithfully  adhered  to, — viz.:  that  the 
memory  should  never  be  trusted.  The  distractions  to 
which  the  mind  is  subject  in  all  such  operations,  necessarily 
call  off*  the  attention,  so  that  a  mere  number,  which  has  no 
associations  to  call  it  up,  will  be  very  likely  to  be  forgotten. 

Perhaps  the  best  method  of  preserving  the  "outs"  is  to 
have  nine  iron  pins  and  five  or  six  brass  ones.  The  leader 
takes  all  the  pins  and  goes  on  until  he  has  exhausted  hia 
iron  pins;  he  then  goes  on  one  chain,  and,  sticking  a 


80  PLANE   TRIGONOMETRY.  [CHAP.  HI. 

brass  pin,  calls,  "  Out."  The  follower  then  advances,  bring- 
ing the  pins.  He  delivers  to  the  leader  the  iron  pins  but 
retains  the  brass  ones.  On  arriving  at  the  end  of  the  line, 
the  brass  pins  in  the  follower's  possession  will  show  the 
number  of  "outs"  and  the  iron  pins  the  number  of  chains 
since  the  last  "out."  Thus,  supposing  he  has  six  brass 
and  eight  iron  pins,  and  that  the  end  of  the  line  is  63  links 
from  the  last  pin,  the  distance  is  68.63  chains. 

149.  Horizontal  Measurement.  In  all  cases  where  the 
object  is  to  determine  the  area  or  the  position  of  points  on 
a  survey,  the  measurements  must  either  be  made  horizon- 
tally, or,  if  made  up  or  down  a  slope,  the  distance  must  be 
reduced  according  to  the  inclination. 

In  chaining  down  a  slope,  the  follower  should  hold  his 
end  of  the  chain  firmly  at  the  pin.  The  leader  should  then 
elevate  his  end  until  the  chain  is  horizontal,  and  then  mark 
the  point  directly  under  the  end  of  the  chain.  This  may  be 
done  by  means  of  a  staff  four  or  five  feet  long,  which  should 
be  held  vertical,  or  by  dropping  a  pin  held  in  the  hand  with 
the  ring  downwards,  or  by  a  plumb-line.  If  the  ground 
slopes  much,  the  whole  chain  cannot  be  used  at  once.  In 
such  cases  the  leader  should  take  the  end  of  the  half  or  the 
quarter,  and,  elevating  it  as  before,  drop  his  pin  or  make  a 
mark.  The  follower  then  comes  forward,  and,  holding  the 
50th  or  25th  link,  as  the  case  may  be,  the  leader  goes  for- 
ward to  the  end  of  another  short  portion  of  the  chain,  which 
he  holds  up,  as  before.  A  pin  is  left  only  at  the  end  of 
every  whole  chain. 

Chaining  up  a  slope  is  less  accurate  than  chaining  down, 
from  the  difficulty  of  holding  the  end  still,  under  the  strain 
to  which  the  chain  is  subjected.  The  follower  should  always, 
in  such  cases,  be  provided  with  a  staff  four  or  five  feet  long, 
and  a  plumb-line  to  keep  it  vertical.  If  the  slope  is  so  steep 
that  the  whole  chain  cannot  be  used  at  once,  the  leader 
should  take  (as  before)  the  end  of  a  short  portion,  say  one 
fourth,  and  proceed  up  hill.  The  follower  then  elevates  his 
end,  holding  it  firmly  against  the  staff,  which  is  kept  vertical 
by  the  plumb-line.  The  leader,  having  made  his  mark,  noti- 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  81 

fies  the  follower,  who  comes  forward  and  holds  up  the 
same  link  that  the  leader  used.  He  then  goes  forward  as 
before. 

150.  When  great  accuracy  is  required,  the  chaining  should 
be  made  according  to  the  slope  of  the  ground,  leaving  stakes 
where  there  is  any  change  of  the  slope,  and  recording  the 
distances  to  these  stakes  in  the  note  book.  The  inclination 
of  the  different  parts  being  then  taken,  the  horizontal  dis- 
tance can  be  calculated.  If  a  transit  with  a  vertical  arc  is 
employed,  the  slope  can  be  obtained  at  once,  and  the  proper 
correction  may  be  made  at  the  time.  The  best  way  is  to  have 
a  table  prepared  for  all  slopes  likely  to  be  met  with,  and 
apply  the  correction  on  the  ground.  Instead  of  deducting 
from  the  distance  measured,  it  is  best  to  increase  the  length 
on  the  slope,  calling  each  length  so  increased  a  chain :  the 
horizontal  distance  will  then  be  correctly  recorded.  Thus, 
supposing  the  slope  to  be  5°,  in  order  that  the  base  may  be 
1  chain  the  hypothenuse  must  be  1.0038 :  the  follower 
should  therefore  advance  his  end  of  the  chain  rather  less 
than  half  a  link. 

If  a  compass  is  used,  it  may  be  furnished  with  a  tangent 
scale,  to  be  described  hereafter. 

The  following  table  contains  the  ratio  of  the  perpen- 
dicular to  the  base,  the  correction  of  the  base  for  each 
chain  on  the  slope,  and  the  correction  of  the  slope  for  each 
horizontal  chain.  If  the  corrections  are  made  as  the  work 
proceeds,  the  last  column  should  be  used ;  if  in  the  field- 
notes  after  the  work  is  done,  the  third  column  furnishes 
the  data. 


82 


PLANE   TRIGONOMETRY. 


[CHAP.  Ill, 


Angle. 

Slope, 
perp.:  base. 

Correction 
of  base,  in 
links. 

Correction 
of  hypoth. 
in  links. 

Angle. 

Slope. 

Correction 
of  base,  in 
links. 

Correction 
of  hypoth. 
in  links. 

3° 

1     19.1 

—0.14 

+0.14 

17° 

1     3.3 

—4.37 

+4.57 

4° 

1     14.3 

0.24 

0.24 

18° 

1     3.1 

4.89 

5.15 

5° 

1     11.4 

0.38 

0.38 

19° 

1     2.9 

6.45 

5.76 

6° 

1       9.5 

0.55 

0.55 

20° 

1     2.7 

6.03 

6.42 

7° 

1       8.1 

0.75 

0.75 

21° 

1     2.6 

6.64 

7.11 

8° 

1      7.1 

0.97 

0.98 

22° 

1     2.5 

7.28 

7.85 

9° 

1      6.3 

1.23 

1.25 

23° 

1    2.4 

7.95 

8.64 

10° 

1      6.7 

1.52 

1.54 

24° 

1     2.2 

8.65 

9.46 

11° 

1      6.1 

1.84 

1.87 

25° 

1    2.1 

9.37 

10.34 

12° 

1      4.7 

2.19 

2.23 

26° 

1     2.1 

10.12 

11.26 

13° 

1      4.3 

2.56 

2.63 

27° 

1    2 

10.90 

12.23 

14° 

1      4.0 

2.97 

3.06 

28° 

1     1.9 

11.71 

13.26 

15° 

1      3.7 

3.41 

3.53 

29° 

1     1.8 

12.54 

14.34 

16° 

1      3.5 

3.87 

4.03 

30° 

1     1.7 

13.40 

15.47 

151.  Tape-Lines.    A  tape-line  is  sometimes  used  instead 
of  a  chain  in  measuring  short  distances.     It  is,  however, 
very  little  to  be  depended  on.     If  used  at  all,  the  kind  that 
is  made  with  a  wire  chain  should  be   employed.     It  is 
much  less  liable  to  be  stretched  than  those  made  wholly 
of  linen. 

152.  Chaining  being  one  of  the  fundamental  operations 
of  surveying,  whether  for  trigonometrical  purposes  or  for 
the   calculation   of   the   contents,   it  has   been    described 
minutely.     If  correct  measurements  are  needful,  accurate 
notes  are  no  less  so.     The  chief  points  to  be  attended  to  in 
recording  the  measurements  are  precision  and  conciseness. 
Some  of  the  most  approved  methods  are  given  in  Chap- 
ter IV. 

153.  Angles.    For  surveying  purposes  horizontal  angles 
alone  are  needed,  since  all  the  parts  of  the  survey  are  re- 
duced to  a  horizontal  plane ;  but  to  fix  the  direction  of  a 
point  in  space  not  only  the  horizontal  but  vertical  angles 
are  required.     With  the  aid  of  these,  and  the  proper  linear 
measures,  its  position  may  be  fully  determined. 

154.  Horizontal  angles  are  measured  by  having  a  plane, 
properly  divided,  and  capable  of  being  so  adjusted  as  to  be 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  '    83 

perfectly  horizontal.  Movable  about  the  centre  of  this 
plane  is  another  plane,  or  a  movable  arm,  carrying  a  pair 
of  sights  or  a  telescope,  which  can  be  placed  so  that  the 
line  of  sight  may  pass  through  the  object.  If  then  this 
line  be  directed  to  one  object,  and  the  position  of  the  two 
plates  or  of  the  arm  on  the  plate  be  noted  by  an  index 
properly  situated,  and  then  be  turned  so  as  to  point  to 
another  object,  the  angle  through  which  the  plate  or  the 
arm  has  turned  will  be  the  horizontal  angle  contained  by 
two  planes  drawn  from  the  centre  of  the  instrument  to  the 
two  objects. 

155.  Vertical  angles  are  measured  by  having  a  pair  of 
sights  or  a  telescope  so  adjusted  as  to  move  on  a  horizontal 
axis,  the  horizontal  position  of  the  sights  or  the  telescope 
being  indicated  either  by  a  plumb-line  or  a  level. 

156.  The  transit  with  a  vertical  arc,  or  the  theodolite, 
are  so  arranged  as  to  perform  both  these  offices.     As  a  full 
understanding  of  the  use  of  the  different  parts  of  these 
instruments  is  necessary  to  their  proper  management,  we 
shall  enter,  considerably  in  detail,  into  a  description  of 
them. 

THE  TRANSIT  AND  THE  THEODOLITE.* 

157.  General  Description.    The  Transit  or  the  Theo- 
dolite (Figs.  51  and  52)  consists  of  a  circular  plate,  divided 
at  its  circumference  into  degrees  and  parts,  and  so  sup- 
ported that  it  can  be  placed  in  a  perfectly  horizontal  posi- 
tion.     This  divided  circle  is  called  the  limb.     An   axis 
exactly  perpendicular  to  this  plate,  bearing  another  cir- 
cular plate,  passes  through  its  centre.     This  plate  is  so 
adjusted  as  to  move  very  nearly  in  contact  with  the  former 
without  touching  it.     By  this  arrangement  the  upper  plate 
can  be  turned   freely  about  their  common   centre.     This 
plate  carries  a  telescope  Q,  resting  on  two  upright  supports 
KK,  upon  which  it  is  movable  in  a  vertical  plane.     The 
telescope,  having  thus  a  horizontal  and  a  vertical  motion, 

*  The  author  is  indebted  to  Professor  Gillespie's  "Treatise  on  Land  Sur- 
veying" for  many  of  the  features  in  his  mode  of  presenting  the  subjects  of  the 
Transit  and  Theodolite,  their  verniers  and  their  adjustments. 


PLANE   TRIGONOMETRY.  [CHAP.  III. 

THE   TRANSIT. 

Fig.  51. 


SEC.V.]  INSTRUMENTS   AND  FIELD  OPERATIONS. 

THE   THEODOLITE. 

Fig.  52. 


85 


86  PLANE  TRIGONOMETRY.  [CHAP.  Ill, 

can  readily  be  pointed  to  any  object.  The  second  described 
plate  has  an  index  of  some  kind,  moving  in  close  proximity 
to  the  divided  arc,  so  that  the  relative  position  of  the  plates 
may  be  determined.  If  then  the  telescope  be  directed  to 
one  object,  and  afterwards  be  turned  to  another,  the  index 
will  travel  over  the  arc  which  measures  the  horizontal  angle 
between  the  objects. 

In  order  to  place  the  plates  in  a  perfectly  horizontal  posi- 
tion, levelling  screws  and  levels  are  required:  these,  as 
well  as  the  other  parts  of  the  instrument,  will  be  fully 
described  in  their  proper  place. 

158.  The  above  description  applies  to  both  instruments. 
The  transit,  however,  is  so  arranged  that  the  telescope  can 
turn  completely  over;   it  can,  therefore,  be  directed  back- 
wards and  forwards  in  the  same  line.     If  the  same  thing  is 
to  be  done  by  the  theodolite,  the  telescope  must  be  taken 
from  its  supports  and  have  its  position  reversed.     This  ope- 
ration is  troublesome,  and  is,  besides,  very  apt  to  derange 
the  position  of  the  instrument. 

For  surveying  purposes,  therefore,  the  transit  is  much  to 
be  preferred;  and  when  the  axis  on  which  the  telescope 
moves  is  provided  with  a  vertical  arc  it  serves  all  the  pur- 
poses of  a  theodolite. 

The  theodolite  has  a  level  attached  to  the  telescope.  This 
is  not  generally  found  in  the  transit. 

159.  The  accuracy  of  these  instruments  depends  on  several 
particulars : — 

1.  By  means  of  the   telescope  the   object  can  be   dis- 
tinctly seen  at  distances  at  which  it  would  be  invisible  by 
the  unassisted  eye. 

2.  The  circle,  with  its  vernier  index,  enables  the  observer 
to  record  the  position  of  the  telescope  with  the  same  degree 
of  precision  with  which  it  can  be  pointed. 

3.  There  are  arrangements  for  giviDg  slow  and  regular 
motion  to  the  parts,  so  as  to  place  the  telescope  precisely  in 
the  position  required. 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  87 

4.  There  are  other  arrangements  for  making  the  plates 
of  the  instrument  truly  horizontal. 

5.  Imperfections  in  the  relative  position  of  the  different 
parts  of  the  instruments  may  be  corrected  by  screws,  the 
heads  of  some  of  which  are  shown  in  the  drawings. 

However  complicated  the  arrangements  for  performing 
these  various  operations  may  make  the  instruments  appear, 
that  complication  disappears  when  they  are  viewed  in  detail 
and  properly  understood. 

160,  In  the  figures  of  these  instruments,  V  is  the  vernier, 
covered  with  a  glass  plate.     In  some  theodolites  the  whole 
divided  limb  is  seen.     In  others  (and  in  the  transit)  but  a 
small  portion  is  exposed, — it  being  completely  covered  by 
the  other  plate,  except  the  small  portions  near  the  vernier. 
Transits  have  generally  but  one  vernier,  though  in  some 
instruments  there  are  two.     The  theodolite  has  generally 
two,  and  sometimes  three  or  four.     B  is  the  compass  box, 
containing  the  magnetic  needle  ET.     A,  A,  are  the  levels. 
C  and  D  are  screws;   the  former  of  which  is  designed  to 
clamp  the  lower  plate,  and  the  latter  to  clamp  the  plates 
together.      T  and  U  are  tangent  screws,  to  give  slow  and 
regular    motion   when  the    plates    are   clamped:    by  the 
former  the  whole  instrument  is  turned  on  its  axis,  and  by 
the  latter  the  upper  plate  is  moved  over  the  other.     P,  P  are 
the  levelling  plates;    and   S,  S,  S,  are   three  of  the  four 
levelling  screws.    E  is  the  vertical  circle,  with  its  vernier  F. 
G  is  a  level  attached  to  the  telescope.     H  is  a  gcrew  to 
clamp  the  horizontal  axis,  (not  visible  in  the  figure  of  the 
theodolite,)   and    I   a   tangent  screw,   to    give    it  regular 
motion. 

161.  The  Telescope.     A  telescope  is  a  combination  of 
lenses  so  adjusted  in  a  tube  as  to  give  a  distinct  view  of  a 
distant  object.     It  consists,  essentially,  of  an  object-glass, 
placed  at  the  far  end  of  the  tube,  and  an  eye-piece  at  the 
near  end. 

By  the  principles  of  optics,  the  rays  of  light  proceeding 
from  the  different  points  of  the  object  are  brought  to  a 


88  PLANE  TRIGONOMETRY.  [CHAP.  III. 

focus  within  the  tube,  (Fig.  53,)  there  forming  an  rig.  53. 
inverted  image.  Crossing  at  this  focus,  they  pro- 
ceed  on  to  the  eye-piece,  by  the  lenses  of  which 
they  are  again  refracted,  and  made  to  issue  in 
parallel  pencils,  thus  giving  a  distinct  magnified 
image  of  the  object. 

162.  The  Object-glass.     Whenever  a  beam  of 
light  passes  through  a  lens,  it  is  not  merely  re- 
fracted, but  it  is  likewise  separated  into  the  different 
colored  rays  of  the  solar  spectrum.     This  separa- 
tion of  the  colored  rays,  or  the  chromatic  aberration, 
causes  the  edges  of  all  bodies  viewed  with  such  a 
glass  to  be  fringed  with  prismatic  colors,  instead  of 
being  sharply  defined.     It  has  been  found,  how- 
ever, that  the  chromatic  aberration  may  be  nearly 

removed,  by  making  a  compound  lens 
of  flint  and  crown  glass,  as  represented  ° 
in  Fig.  54,  in  which  A  is  a  concavo- 
convex  lens  of  flint  glass,  and  B  a 
double  convex  lens  of  crown  glass, — the 
convexity  of  one  surface  being  made  to 
agree  with  the  concavity  of  the  other 
lens.  The  two  are  pressed  together  by  a  screw 
in  the  rim  of  the  brass  box  which  contains  them, 
thus  forming  a  single  compound  lens.  When  the 
surfaces  are  properly  curved,  this  arrangement  is 
nearly  achromatic. 

The    object-glass   is  placed    in   a   short    tube, 
movable  by  a  pinion  attached  to  the  milled  head 
W.  (Figs.  51, 52.)    By  this  means  it  may  be  moved 
backwards  and  forwards,  so  as  to  adjust  it  to  dis-  B 
tinct  vision. 

c 

163.  The  Eye-piece.     The  eye-piece  used  in 
the   telescopes  employed  for  surveying  purposes 
consists  of  two   plano-convex  lenses,  fixed  in  a 
short  tube,  the  convex  surfaces  of  the  lenses  being  A 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  89 

towards    each    other.      This    arrangement    is    known    as 
"  Eamsden's  Eye-piece." 

164.  A  telescope  with  an  object-glass  and  an  eye-piece 
as  above  described,  inverts  objects.     By  the  addition  of  two 
more   lenses   the  rays  may  be  made  to  cross   each  other 
again,  and  thus  to  give  a  direct  image  of  the  object.     As 
these  additional  lenses  absorb  a  portion  of  the  light  passing 
through  them,  they  diminish  the  brightness  of  the  image. 
They  may  therefore  be  considered  a  defect  in  telescopes 
intended  for  the  transit  or  theodolite.      A  little  practice 
obviates  the  inconvenience  arising  from  the  inversion  of 
the  image.     The  surveyor  soon  learns  to  direct  his  assistant 
to  the  right  when  the   image   appears   to  the  left  of  its 
proper  position,  and  vice  versa. 

165,  The  Spider-Lines.    The  advantage  gained  by  the 
telescope  in  producing  distinct  vision,  would  add  nothing  to 
the  precision  of  the  observations,  without  some  means  of 
directing  the  attention  to  the  precise  point  which  should 
be  observed  in  the  field  of  view.     The  whole  field  forms  a 
circle,  in  the  centre  of  which  the  object  should  appear  at 
the  time  its  position  is  to  be  noted.     This  centre  is  de- 
termined by  stretching  across  the  field  precisely  in  the  focus 
of  the  eye-piece  a  couple  of  spider-lines  or  fine  wires,  at 
right  angles   to   each   other.      The   former   are  generally 
employed.     "When  they  are  properly  adjusted  in  the  focus 
they  can  be  distinctly  seen,  and  the  point  to  be  observed 
can  be  brought  exactly  to  coincide  with  their  intersection. 
The  magnifying  power  of  the  eye-piece  enables  this  to  be 
done  with   the    greatest    precision.      When  it  has    been 
effected,  a  line  through  the  centre  of  the  eye-piece  and  the 
centre  of  the  object-glass  will  pass  directly  through  the 
object.     This  line  is  called  the  line  of  collimation  of  the 
telescope. 

The  spider-lines  are  attached  by  gum  to  the  rim  of  a 
circular  ring  of  brass  placed  in  the  tube  of  the  telescope  at 
the  point  indicated  by  the  screw-heads  a,  a,  (Figs.  51,  52,) 
eome  of  which  are  invisible  in  the  figure.  These  screws 


90  PLANE   TRIGONOMETRY.  [CHAP.  III. 

serve  to  hold  the  ring  in  position,  as  Fis-  55- 

represented  in  Fig  55,  and  to  adjust  - 
it  to  its  proper  position.  The  eye- 
piece is  made  to  slip  in  and  out  of  the 
tube  of  the  telescope,  so  that  the  focus 
may  be  brought  to  coincide  exactly 
with  the  intersection  of  cross-wires. 
The  perfect  adjustment  of  the  focus 
may  be  determined  by  moving  the 
eye  sideways.  If  this  motion  causes  the  wires  to  change 
their  position  on  the  object,  the  adjustment  is  not  perfect: 
it  must  be  made  so  before  taking  the  observation. 

166.  Spider-lines  are  generally  used  for  making  the 
"cross- wires,"  though  platinum  wires  drawn  out  very  fine 
are  preferable.  The  wire  is  drawn  to  the  requisite  degree 
of  fineness  by  stretching  a  platinum  wire  in  the  axis  of  a 
cylindrical  mould  and  casting  silver  around  it.  The  com- 
pound wire  thus  formed  is  then  drawn  out  as  fine  as  possi- 
ble and  the  silver  removed  by  nitric  acid.  By  this  means 
Dr.  Wollaston  succeeded  in  obtaining  wire  not  more  than 
one  thirty  thousandth  (^fan)  of  an  inch  in  diameter.  As 
such  wire  is  very  difficult  to  procure,  the  spider-threads  are 
generally  substituted.  The  operation  of  placing  them  in 
their  proper  position  is  thus  performed.  A  piece  of  stout 
wire  is  bent  into  the  form  of  the  letter  U,  the  distance  between 
the  legs  being  greater  than  the  external  diameter  of  the 
ring.  A  cobweb  is  selected  having  a  spider  hanging  at  the 
end.  It  is  gradually  wound  round  the  wire,  his  weight 
keeping  it  stretched :  a  number  of  strands  are  thus  obtained 
extending  from  leg  to  leg  of  the  wire :  these  are  fixeji  by  a 
little  gum. 

To  fix  them  in  their  position,  the  wire  is  placed  so  that 
one  of  the  lines  is  over  notches  previously  made  in  the  ring. 
The  thread  is  then  fixed  in  the  position  with  gum  or  some 
other  tenacious  substance.  The  wire  being  removed,  the 
line  is  left  stretched  across  the  opening  in  the  proper 
position. 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  91 

167.  The  Supports.     Attached  to  one  of  the  horizontal 
plates,  usually  the  index-plate  of  the  instrument,  are  two 
supports,  K,  K,  (Figs.  51,  52,)  bearing  the  horizontal  axis 
L.     These  supports  should  be  made  of  precisely  the  same 
height,  so  that  when  the  plate  is  level  the  axis  may  be  hori- 
zontal.    In  some  instruments  there  is  an  arrangement  for 
raising  or  depressing  one  end  of  the  axis  so  as  to  perfect 
the  adjustment.     In  most  cases,  however,  the  adjustment 
is  made  perfect  by  the  maker,  and,  if  found  not  to  be  so,  it 
"rnust  be  remedied  by  removing  the  support  which  is  too 
high  and  filing  some  off  from  the  bottom.     This  should 
always  be  done  by  the  manufacturer, 

In  the  transit  the  telescope  is  attached  immediately  to  the 
axis ;  but  in  the  theodolite  the  axis  bears  a  bar  M  at  right 
angles  to  it.  This  bar  carries  at  its  ends  two  supports,  which 
from  their  shape  are  called  Y's,  in  the  crotch  of  which  the 
telescope  rests,  being  confined  there  by  an  arch  of  metal 
passing  over  the  top.  This  arch  is  movable  by  a  joint  at 
one  side,  and  is  fastened  by  a  pin  at  the  other.  By  remov- 
ing the  pin  and  lifting  the  arch  the  telescope  is  released  and 
may  be  taken  from  the  support.  It  rotates  freely  on  its  axis 
when  confined  by  the  arch.  The  telescope,  being  attached 
thus  to  the  horizontal  axis,  admits  of  being  elevated  or 
depressed  in  a  vertical  plane  so  that  it  may  be  directed  to 
any  object. 

168.  The  Vertical  Limb.     In  the  theodolite,  the  vertical 
limb  E  consists  of  a  semicircle  of  brass  graduated  on  its 
face  and  attached  to  the  bar  M.     This  limb  moves  with  the 
telescope  upon  the  horizontal  axis,  and  thus  by  means  of 
the  index  F,  serves  to  determine  the  angle  of  elevation  of 
the  object.     In  the  transit  with  a  vertical  circle,  the  circle 
is  attached  to  the  end  of  the  axis,  as  seen  at  E,  the  index 
then  being  attached  to  the  support  K.     In  some  instru- 
ments, instead  of  the  axis  bearing  a  circle,  an  arc  of  from 
60°  to  90°  is  attached  to  the  support,  and  the  index  is  fixed 
to  the  axis  by  an  arm  which  is  either  permanently  fastened 
to  it  or  is  capable  of  being  clamped  in  any  position. 


92  PLANE   TRIGONOMETRY.  [CHAP.  IIL 

169.  The  Levels.     Attached  to  the  horizontal  plate  are 
two  levels  A  and  A  set  at  right  angles  to  each  other,  so  as 
to  determine  when  that  plate  is  horizontal.     They  consist 
of  glass  tubes  very  slightly  curved,  the   convexity  being 
upward.    They  are  nearly  filled  with  alcohol,  leaving  a  small 
bubble  of  air,  which  by  the  principles  of  hydrostatics  will 
always  take  the  highest  point.    If  they  are  properly  adjusted, 
the  plate  to  which  they  are  attached  will,  when  these  bub- 
bles have  been  brought  to  the  middle  of  their  run,  be  level, 
however  it  may  be  turned  about  its  vertical  axis.     To  the 
telescope  of  the  theodolite  and  also  to  that  of  some  transits 
another  level  G  is  fixed.     This  should  be  so  adjusted  that 
when  the  line  of  collimation  of  the  telescope  is  horizontal 
the  bubble  may  be  in  the  centre  of  its  run. 

170.  The  Levelling  Plates.    The  four  screws  S,  S,  S,  and 
S,  called  levelling  screws,  are  arranged  at  intervals  of  90° 
between   the  two  plates  P,  P,  which  are  called  levelling 
plates  or  parallel  plates.     They  screw  into  one  plate  and 
press  on  the  other.     By  tightening  one  screw  and  loosening 
the  opposite  one  at  the  same  time,  the  upper  plate,  with  the 
instrument  above,  may  be  tilted.     To  allow  this  motion,  the 
column  connecting  them  terminates  in  a  ball,  which  works 
in  a  socket  in  the  centre  of  the  lower  plate.     A  joint  of  this 
kind,  called  a  ball-and-socket  joint,  allows  movement  in  all 
directions. 

To  level  the  instrument  by  means  of  these  levelling 
screws,  loosen  the  clamp,  and  turn  the  plates  until  the 
telescope  is  directly  over  one  pair  of  the  screws.  Then, 
taking  hold  of  two  opposite  screws,  move  them  in  contrary 
directions  with  an  equal  and  uniform  motion,  until  the 
bubble  in  the  tube  parallel  to  the  line  joining  these  screws 
is  in  the  middle.  Then  turn  the  other  screws  in  like  man- 
ner until  the  other  bubble  comes  to  the  middle  of  its  tube. 
When  they  are  both  brought  to  this  position  the  plates  are 
level  if  the  instrument  is  in  adjustment.  In  levelling,  care 
should  be  taken  to  move  both  screws  equally.  If  one  is 
moved  faster  than  the  other,  the  instrument  will  not  be  firm, 
or  will  be  cramped. 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  93 

171.  The  Clamp  and  Tangent  Screws.    The  former  of 
these  are  used  for  binding  parts  of  the  instrument  firmly 
together,  the  latter  for  giving  a  slow  motion  when  they  are 
so  bound.      The  clamp  C  tightens  the   collar  0  clasping 
the  vertical  axis,  and  thus  holds  it  and  the  plate  attached 
to  it  firmly  in  their  places.     The  other  plate,  moving  on  an 
axis  within  the  former,  may,  notwithstanding,  move  freely. 
"When  this  clamp  is  tightened,  the  collar  may  be  moved 
slowly  round  by  means  of  the  tangent  screw  T.     In  its 
motion  it  carries  with  it  the  axis  and  attached  plate.     The 
clamp  D  fastens  the  two  plates  together.     They  may,  how- 
ever, when  so  clamped,  be  made  to  move  slightly  on  each 
other  by  means  of  the  tangent  screw  U.      If  both  clamps 
are  tight,  the  instrument  is  firm,  and  the  telescope  can  only 
be  turned  horizontally  by  one  of  the  tangent  screws.     If 
the  clamp  C  is  tight  and  the  other  loose,  the  telescope  and 
upper  plate  will  move  while  the  lower  remains  fixed.     If  D 
is  tight  and  C  is  loose,  the  two  plates  are  firmly  attached 
to  each  other;   but  the  whole  instrument  can  be  moved 
horizontally. 

Attached  to  the  horizontal  axis  there  is  likewise  a  clamp 
H  and  tangent  screw  I,  the  purposes  of  which  are  similar 
to  those  described, — the  clamp  fixing  the  axis,  and  the 
screw  moving  it  slowly  and  steadily. 

172.  The  Watch-Telescope.    Connected  with  the  lower 
part  of  theodolites  of  the  larger  class  there  is  a  second  tele- 
scope B,,  the  object  of  which  is  to  determine  whether  the  in- 
strument has  changed  position  during  an  observation.     It 
is  directed  to  some  well  defined  object,  and  after  all  the  ob- 
servations at  the  station  have  been  made,  or  more  frequently 
if  thought  necessary,  it  should  be  examined  to  see  whether 
or  not  it  has  changed  its  position.     If  it  has,  the  divided 
arc  has  changed  also.     The  instrument,  therefore,  requires 

readjustment. 

• 

173.  Verniers.    As  it  would  be  very  difficult  to  divide  a 
circle  to  the  degree  of  minuteness  to  which  it  is  desirable 
to  read  the  angles,  or,  if  it  were  so  divided,  since  it  would 


94 


PLANE   TRIGONOMETRY. 


[CHAP.  III. 


be  impossible  for  the  eye  to  detect  the  divisions,  some 
contrivance  is  necessary  to  avoid  both  difficulties.  These 
difficulties  will,  perhaps,  be  made  more  striking  by  a 
simple  calculation.  The  circumference  of  a  circle  6  inches 
in  diameter  is  18.849  inches.  If  the  circle  is  divided  into 


degrees  there  will  be  --  =19.1  divisions  in  the  space 

of  an  inch.  If  the  divisions  are  quarter  degrees  there  will 
be  76.4  to  the  inch  ;  and  if  minutes,  there  would  be  1150 
divisions  to  every  inch.  The  first  and  second  could  be 
read  ;  but  the  third,  though  it  might  by  proper  mechanical 
contrivances  be  made,  yet  it  would  be  almost,  if  not  en- 
tirely, impossible  to  distinguish  the  cuts  so  as  to  read  the 
proper  arc.  And  yet  that  division  is  not  so  minute  as  is 
sometimes  desirable  on  a  circle  of  that  diameter.  The 
vernier  is  a  simple  contrivance  to  effect  this  subdivision  of 
space,  in  a  way  to  be  perfectly  distinct  and  easily  read. 

174.  The  principle  of  the  vernier  will  be  best  understood 
by  a  simple  example.  In  the  adjoining  figure,  (Fig.  56,) 
AB  represents  a  scale  with  the  inch  divided  into  tenths,  the 
figure  being  on  a  scale  of  3  to  2  or  1J  times  the  natural  size. 

Fig.  56. 


/ 

\ 

10 

5                            ° 

3 

0 

2 

9 

2 

CD  is  another  scale  having  a  space  equal  nine  of  the 
divisions  on  AB  divided  into  ten  equal  parts.  This  second 
scale  is  the  vernier.  Kow,  since  ten  spaces  of  the  vernier  are 
equal  to  nine  of  the  scale,  each  of  the  former  is  equal  to  nine 
tenths  of  one  of  the  latter.  If  then  the  0  on  the  vernier 
corresponds  to  one  of  the  divisions  of  the  scale,  the  first 
division  of  thervernier  will  fall  ^  of  a  space  or  ^  of  an 
inch  below  the  next  mark  on  the  scale,  the  next  -division 


SEC.  V.] 


INSTRUMENTS  AND  FIELD  OPERATIONS. 


95 
and  so  on.     The 


will  fall  ifo  °f  an  mcn  below,  the  next  __ 
0  in  the  figure  stands  at  28.7  inches. 

If  now  the  vernier  be  slid  up  so  that  the  first  division 
shall  correspond  to  a  division  on  the  scale,  the  0  will  have 
been  raised  ^  inch.  If  the  second  be  made  to  coincide, 
the  vernier  will  have  been  raised  ^  of  an  inch.  If  it  be 
placed  as  in  Fig.  57,  the  reading  will  be  28.74  inches. 

Fig.  57. 


s  o 


•2  S 


The  student  should  make  for  himself  paper  scales,  di- 
vided variously,  with  verniers  on  other  pieces  of  paper,  so 
that  he  may  become  familiar  with  the  manner  of  reading 
them.  If  his  scale  is  to  represent  degrees,  the  portion  re- 
presenting the  arc  might  be  drawn  as  a  straight  line,  for  the 
sake  of  facility  in  the  drawing.  It  will  illustrate  the  subject 
as  well  as  if  an  arc  of  a  circle  were  used.  He  should  be- 
come particularly  familiar  with  the  one  represented  by  Fig. 
60,  as  it  is  the  division  most  commonly  used  in  theodolites 
and  transits. 

175.  The  Beading  of  the  Vernier.  To  determine  the 
reading  of  the  vernier, — that  is,  the  denomination  of  the  parts 
into  which  it  divides  the  spaces  on  the  scale, — obsj^rvejiow 
many  of  the  spaces  on  the  scale  are  equal  to  a  number  on 
the  vernier  which  is  greater  or  less  by  one.  The  number  of 
spaces  on  the  vernier,  so  determined,  divided  into  the  value 
of  one  of  the  spaces  on  the  scale,  will  give  the  denomination 
required.  Thus,  in  Figs.  56  and  57,  ten  spaces  of  the  ver- 
nier correspond  with  niae  on  the  scale :  the  reading  is 
therefore  to  fa  of  fa  =  ^  of  an  inch. 

If  an  arc  were  divided  into  half-degrees,  and  thirty  spaces 
on  the  vernier  were  equal  to  twenty  nine  or  to  thirty  one 


36  PLANE   TRIGONOMETRY.  [CHAP.  IIL 

spaces  on  the  arc,  the  reading  would  be  to  ^  of  |°  =  ^5°  =  1 
minute ;  or,  as  it  is  usually  expressed,  to  minutes.  Fig.  60 
is  an  example  of  this  division. 

\  176.  To  read  any  Vernier.  First,  determine  as  above 
the  reading.  Then  examine  the  zero  point  of  the  vernier. 
If  it  coincides  with  any  division  of  the  scale  as  in  Fig.  56, 
that  division  gives  the  true  reading, — 28.7  inches.  But  if, 
as  will  generally  be  the  case,  it  does  not  so  coincide,  note  the 
division  of  the  scale  next  preceding  the  place  of  the  zero,  and 
then  look  along  the  vernier  until  a  division  thereof  is  found 
which  is  in  the  same  straight  line  as  some  division  on  the 
scale.  This  division  of  the  vernier  gives  the  number  of 
parts  to  be  added  to  the  quantity  first  taken  out.  Thus,  in 
Fig.  57,  the  0  of  the  vernier  is  between  8.7  and  8.8,  and 
the  fourth  division  on  the  vernier  is  in  a  line  with  a  division 
on  the  scale :  the  true  reading  is  therefore  28.74  inches. 

To  assist  the  eye  in  determining  the  coincidence  of  the 
lines,  a  magnifying  glass,  or  sometimes  a  compound  micro- 
scope, is  employed. 

When  no  line  is  found  exactly  to  coincide,  then  there  will 
be  some  which  will  appear  equally  distant  on  opposite  sides. 
In  such  cases,  take  the  middle  one. 

177.  Retrograde  Verniers.  Most  verniers  to  modern 
instruments  are  made  as  above  described.  In  some  in- 
stances, the  vernier  is  made  to  correspond  to  a  number  of 
spaces  on  the  arc  one  greater  than  that  into  which  it  is 
divided.  Such  verniers  require  to  be  read  backwards,  and 
are  hence  called  retrograde  verniers.  Fig.  58  is  an  ex- 
ample of  one  of  this  kind.  It  is  the  form  that  is  generally 
used  in  barometers.  It  is  drawn  to  one  and  a  half  times  the 
natural  size :  the  inches  are  divided  into  tenths,  and  eleven 
spaces  on  the  scale  correspond  with  ten  on  the  vernier. 


SEC.  V.]  INSTRUMENTS  AND   FIELD   OPERATIONS. 

Fig.  58.  - 


97 


0 

1 

> 

1 

0 



1 

1    1 

| 

f 

| 

3 

0 

2 

9 

2 

8 

The  value  of  one  division  of  the  vernier  is  T^ inch.  If 
therefore  0  on  the  vernier  corresponds  to  a  division  on  the 
scale,  1  on  the  vernier  will  be  ^  of  an  inch  below  the  next 
on  the  scale,  2  will  be  yfo  below ;  and  so  on.  If  the  vernier 
is  raised  so  that  the  1  on  the  vernier  is  in  line,  it  is  raised 
^  inch;  if  2  is  in  line,  it  is  raised  jfo;  and  so  on.  The 
reading  in  Fig.  58  is  29.7  inches,  and  in  Fig.  59,  29.53 
inches. 

Fig.  59. 


0 

V 

5 

l 

0 

! 

1 

1 

a 

0 

2 

9 

2 

8 

178.  In  Fig.  60,  the  arc  is  divided  by  the  longer  lines  into 
degrees,  and  by  the  shorter  into  half  degrees,  or  30'  spaces. 


Fig.  60. 


98 


PLANE   TRIGONOMETRY. 


[CHAP.  III. 


Thirty  spaces  on  the  vernier  are  equal  to  twenty  nine  on 
the  arc.  The  reading  is  therefore  to  ^  of  30  minutes  =  1 
minute.  The  zero  of  the  vernier  stands  between  41°  30' 
and  42°.  On  looking  along  the  vernier,  it  is  seen  that  the 
fifth  and  sixth  lines  coincide  about  equally  well.  The  ver- 
nier therefore  reads  41°  35'  30" 

179.  Reading  backwards.  Sometimes  it  is  required  to 
read  backwards  from  the  zero  point  on  the  limb.  When 
this  is  done,  the  numbers  on  the  vernier  must  be  read  in 
reverse,  the  highest  being  called  zero,  and  the  zero  the 
highest. 

Fig.  61. 


Thus,  in  Fig.  61,  the  zero  of  the  vernier  standing  to  the 
right  of  360  on  the  limb,  between  1°  30'  and  2°,  and  the 
division  marked  with  an  arrow-head  being  in  line,  the  angle 
is  1°  41'.  This  mode  of  reading  is  needful  when  using  the 
theodolite  to  take  angles  of  depression,  and  also  when  using 
the  transit  to  trace  a  line  that  bends  backwards  and  for- 
wards, the  angle  of  deflection  being  then  generally  taken, 
and  recorded  to  the  right  or  to  the  left,  as  the  case  may  be. 

180.  Double  Verniers.  To  avoid  the  inconvenience  of 
reading  backwards,  a  double  vernier  is  frequently  made.  It 
consists  of  two  direct  verniers  having  the  same  zero  point, 
as  shown  in  Fig.  62. 


SEC.  V.]  INSTRUMENTS  AND   FIELD   OPERATIONS. 

Fig.  62. 


The  arc  in  this  figure  is  divided  into  degrees,  and  eleven 
spaces  on  the  arc  are  equal  to  twelve  on  the  vernier :  the 
reading  is  therefore  to  5  minutes.  When  the  figures  "on  the 
arc  increase  to  the  right,  the  right-hand  vernier  is  used,  and 
vice  versa.  The  reading  on  the-  figure  is  2°  45'  to  the  left. 

181.  Another  form  of  double  vernier  is  shown  in  Fig.  63. 


.  63. 


In  the  figure,  the  vernier  reads  to  minutes.  When  the 
zero  of  the  vernier  is  to  the  left  of  that  on  the  limb,  the 
figures  begin  at  the  zero  and  increase  towards  the  left  to 
15' ;  they  then  pass  to  the  right-hand  extremity,  and  again 
proceed  to  the  left ;  that  is,  they  stop  at  A  and  commence 
again  at  B.  The  upper  figures  of  each  half  are  the  con- 
tinuation of  the  lower  figures  of  the  other  half.  The  read- 
ing in  Fig.  63  is  1°  8'  to  the  left. 

In  Fig.  64  the  reading  is  3°  19'  to  the  right. 


100 


PLANE  TRIGONOMETRY. 

Fig.  64. 


[CHAP.  IIL 


Fig.  65. 
E 


182.  If  the  preceding  descriptions  have  been  thoroughly 
understood,  the  student  will  have  no  difficulty  in  reading 
the  arc  on  any  limb,  however  it  may  be  divided.  He  should 
study  the  different  positions  until  he  can  determine  the 
angle  with  readiness,  however  the  index  may  be  placed. 
For  this  purpose,  as  before  remarked,  he  should  make  for 
himself  verniers  with  different  scales,  so  that  they  can  be 
placed  in  various  positions. 

The  construction  of  such  verniers  is  very  simple.  Suppose, 
for  example,  it  is  desired  to  divide  the  arc  into  degrees  and 
subdivide  it  by  the  vernier  so  as  to  read  to  5  minutes :  twelve 
spaces  on  the  vernier  must  equal  eleven  on  the  arc,  or  one 
space  on  the  vernier  will  equal  JJ  of  a  space  on  the  arc.  Let 
(Fig.  65)  E  be  the  centre  and  AB  a  por- 
tion of  the  limb,  which,  for  the  purpose 
intended,  should  not  be  of  less  radius 
than  ten  or  twelve  inches,  and  let  CD  be 
the  vernier;  with  some  other  radius  EG, 
which  should  be  greater  than  EB,  de- 
scribe an  arc  GF;  take  El :  EG : :  number  A 
of  divisions  on  the  vernier  :  the  number 
that  occupies  the  same  space  on  the  arc,  H 
— in  this  case,  as  12  to  11.  Take  from 
the  table  of  chords  the  chord  of  1°  or  £°,  as  the  case  may  be, 
and  multiply  it  by  the  length  of  EG ;  lay  off  the  product  on 
GF,  thus  determining  the  points  1,  2,  3,  &c.,  and  lay  off  the 
same  length  on  IH,  determining  the  points  a,  6,  c,  &c. ;  stick 
a  fine  needle  in  the  centre  E;  then,  resting  the  ruler  against 
the  needle,  bring  it  so  as  to  coincide  with  I,  and  draw  the 


SEC.  V.]  INSTRUMENTS   AND  FIEL1)  OPERATIONS.  101 

division  on  AB ;  then,  keeping  it  pressed  against  the  needle, 
bring  it  successively  to  the  other  points  on  GF,  and  draw 
the  corresponding  divisions  on  AB.  The  arc  will  then  be 
divided.  In  the  same  way,  resting  the  ruler  against  the 
needle,  and  bringing  it  successively  to  the  points  on  IH,  the 
vernier  may  be  divided.  The  reason  of  this  process  is,  that 
since  ab  —  1.2,  the  degrees  of  ab  will  be  to  the  degrees  of 
1.2  as  the  radius  of  GF  is  to  the  radius  of  HI,  as  11  to  12. 
Hence  each  division  of  the  vernier  is  $  of  one  division  of 
the  arc. 

By  this  means  the  divisions  may  be  made  with  facility 
and  accuracy. 

183.  Adjustments.    In  order  that  the  theodolite  and 
transit  may  give  correct  results  when  used,  it  is  necessary 
that  the  different  parts  should  bear  the  precise  relations  to 
each  other  that  they  are  intended  to  have.    By  the  term 
adjustment  is  meant  the  due  relation  of  the  parts  to  each 
other :  when  it  is  said  an  instrument  is  in  adjustment,  it  is 
meant  that  every  part  bears  to  every  other  precisely  its 
proper  relations,  so  that  the  instrument  is  in  perfect  work- 
ing order. 

Before  making  any  observations  with  a  new  instrument, 
it  should  be  carefully  examined  to  verify  the  adjustment. 
If  the  parts  are  not  found  to  be  properly  adjusted,  they 
must  be  rectified. 

184.  For  measuring  horizontal  angles,  the  following  con- 
ditions are  necessary  :— 

1.  The  levels  should  be  parallel  to  the  plates,  so  that 
when  the  bubbles  are  in  the  middle  of  their  run,  the  plates 
shall  be  horizontal. 

2.  The  axes  of  the  two  horizontal  plates  should  be  per- 
fectly parallel  and  perpendicular  to  the  plane  of  the  plates. 

3.  The  line  of  collimation  should  be  perpendicular  to  the 
horizontal  axis. 

4.  The  horizontal  axis  should  be  parallel  to  the  plane  of 
the  plates,  so  that  when  they  are  horizontal  it  may  be  so 
likewise. 


102  "  PLANE   TRIGONOMETRY.  [CHAP.  III. 

185.  First  Adjustment.  The  levels  should  be  parallel  to 
the  horizontal  plates. 

Verification.  Clamp  the  two  plates  together;  loosen  the 
clamp  C,  (Figs.  51,  52 ;)  bring  the  telescope  directly  over  one 
pair  of  levelling  screws,  and  level  the  plates  as  directed  in 
Art.  170.  Turn  the  plates  half  round :  if  the  bubbles  retain 
their  position,  the  plane  of  the  levels  is  perpendicular  to 
the  axis  on  which  the  lower  plate  turns.  If  either  of  them 
inclines  to  one  end  of  its  tube,  it  is  out  of  adjustment,  and 
requires  rectification. 

To  rectify  the  fault,  bring  the  bubble  halfway  back  to  the 
middle  by  means  of  the  capstan  screw  attached  to  one  end, 
and  the  other  half  by  the  levelling  screws.  Again  reverse 
the  position  of  the  plate  :  if  the  bubble  now  remains  in  the 
middle,  the  rectification  is  complete ;  if  not,  the  operation 
must  be  repeated.  When  both  levels  have  been  so  arranged 
that  the  bubbles  retain  their  position  in  the  middle  of  their 
run  when  the  plates  are  turned  all  round,  the  adjustment 
is  perfect,  and  the  axis  is  perpendicular  to  the  plane  of  the 
levels. 


186.  Second  Adjustment.  The  axes  of  the  horizontal 
plates  should  be  parallel. 

Verification.  Level  the  plates,  as  directed  in  last  article. 
Clamp  the  lower  plate,  and  loosen  the  vernier-plate.  Turn 
it  half  round :  if  both  bubbles  still  retain  their  position  the 
axes  are  parallel.  If  the  plates  move  freely  over  each 
other  without  binding  in  any  position,  they  are  perpendi- 
cular to  the  axes,  or,  at  least,  the  upper  one  is  so. 

If  any  defects  be  found  in  either  of  these  particulars, 
the  instrument  should  be  returned  to  the  maker  to  be 
rectified. 


187.  Third  Adjustment.  The  line  of  collimation  of  the 
telescope  of  the  theodolite  should  be  parallel  to  the  common  axis 
of  the  cylinders  on  which  it  rests  in  its  Y  '&• 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  1Q3 

Verification.  Direct  the  telescope  so  that  the  intersec- 
tion of  the  wires  bisects  some  well  defined  point  at  a  dis- 
tance. Rotate  the  telescope  so  as  to  bring  the  level  to  the 
top.  If  the  intersection  still  coincides  with  the  object, 
the  adjustment  is  perfect.  If  it  has  changed  its  posi- 
tion, bring  it  half-way  back,  b}r  the  screws  a,  a,  and  verify 
again. 


188.  Fourth.  Adjustment.     The  Une  of  collimation  must  be 
perpendicular  to  the  horizontal  axis. 

Verification  for  the  Transit.  Set  the  transit  on  a  piece 
of  level  ground,  as  at  A,  (Fig.  66,)  and  level  it  carefully. 
At  some  distance — say  four  or  five  chains — set  a  stake  B 
in  the  ground,  with  a  nail  driven  in  the  head,  and  direct 
the  telescope  so  that  the  cross-  c  Fis- 66- 

wires  may  bisect  exactly  on  the 
nail.  Clamp  the  plates,  turn 
the  telescope '  over,  and  place  a 
second  stake  C  precisely  in  the 

line  of  sight.  If  the  adjustment  is  perfect,  the  three  points 
B,  A,  and  C  will  be  in  a  straight  line.  To  determine 
whether  they  are  so,  turn  the  plate  round  until  the  tele- 
scope points  to  B ;  turn  it  over,  and,  if  the  line  of  sight 
passes  again  through  C,  the  adjustment  is  perfect.  If  it 
does  not,  set  up  a  stake  at  E,  in  the  line  of  sight :  then  the 
prolongation  of  the  line  BA  bisects  EAC. 

Let  FG  (Fig.  67)  be  the 
horizontal  axis.  Then,  if 
the  line  of  collimation 
makes  the  angle  FAB 
acute,  when  the  telescope  D 
is  turned  over  it  will  make 
FAC  =  FAB.  The  angle 
CAD  is  therefore  equal  to 
twice  the  error.  Now,  if  the  plate  is  turned  until  the  line 
of  sight  is  directed  to  B,  the  axis  will  be  in  the  position 
F'G'.  Turn  the  telescope  over,  and  the  angle  EAF'= 
F'AB ;  CAE  is  therefore  equal  to  four  times  the  error. 


104  PLANE  TRIGONOMETRY.  [CHAP.  III. 

Hence,  to  rectify  the  error,  the  instrument  being  in  the 
second  position,  place  a  stake  at  H,  one  fourth  of  the  dis- 
tance from  E  to  C,  (Fig.  67,)  and,  by  means  of  the  screws 
<z,  «,  (Fig.  51,)  move  the  diaphragm  horizontally  till  the 
vertical  line  passes  through  H.  Verify  the  adjustment; 
and,  if  not  precisely  correct,  repeat  the  operation. 


189.  The  above  method  is  inapplicable  to  the  theodolite, 
as  its  telescope  does  not  turn  over.  For  the  means  of 
detecting  and  correcting  the  error,  see  Art.  190. 


190,  Fifth  Adjustment.  The  horizontal  and  the  vertical 
axes  should  be  perpendicular. 

Verification  for  the  Transit.  Suspend  a  long  plumb-line 
from  some  elevated  point,  allowing  the  plummet  to  swing 
in  a  bucket  of  water ;  then  level  carefully,  and  bisect  the 
line  accurately  by  the  vertical  wire.  If,  on  elevating  and 
depressing  the  telescope,  the  line  is  still  bisected,  the  ad- 
justment is  good.  If  not,  the  error  may  be  corrected  by 
filing  one  of  the  frames.  Instead  of  a  plumb-line,  any  ele- 
vated object  and  its  image,  as  seen  reflected  from  the  surface 
of  mercury  or  of  molasses  boiled  to  free  it  from  bubbles, 
may  be  employed. 

Verification  for  the  Iheodolite.  If  the  instrument,  treated 
as  above,  shows  a  defect,  the  error  may  be  either  in  the 
axis,  or  in  the  position  of  the  Y's.  To  determine  which, 
turn  the  plates  half  round,  and  reverse  the  telescope.  If 
the  deviation  is  now  on  the  same  side  as  before,  the  Y's  are 
in  fault.  Their  position  in  most  instruments  may  be  cor- 
rected by  screws  which  move  one  of  them  laterally.  If  the 
line  deviates  to  the  opposite  side  from  before,  the  position 
of  the  axis  may  be  corrected  by  filing,  as  directed  for  the 
transit. 

This  adjustment  may  also  be  examined  by  directing  the 
telescope  to  some  well  defined  elevated  object,  and  then  to 


SEC.  V.] 


INSTRUMENTS  AND  FIELD  OPERATIONS. 


105 


another  on  or  near  the  ground.  If  none  such  can  be 
found,  let  one  be  placed  by  an  assistant ;  then  reverse  the 
telescope  in  its  Y's  if  the  instrument  is  a  theodolite,  or 
turn  it  over  if  the  instrument  is  a  transit,  and  direct  it  to 
the  upper  object.  If  the  cross- wires  still  intersect  upon 
the  lower  point  when  the  tube  is  depressed,  the  adjustment 
is  perfect. 

191.  Adjustments  of  the  Vertical  Limb.     Having 
verified  the  various  adjustments  for  horizontal  motion,  as 
described  in  the  preceding  articles,  and  rectified  them  if 
defective,  the  instrument  is  ready  for  use  for  horizontal 
work.     To  take  angles  of  elevation,  or  to  use  the  instru- 
ment for  levelling,  the  following  adjustments  must  also  be 
examined : — 

1.  The  level  beneath  the  telescope  must  be  parallel  to  the 
line  of  collimation. 

2.  The  zero  of  the  vernier  must  coincide  with  the  zero 
of  the  vertical  limb  when  the  plates  are  level  and  the  tele- 
scope horizontal. 

192.  First  Adjustment.     The  level  must  be  parallel  to  the 
line  of  collimation. 

Verification.  Select  a  piece  of  level  ground,  and  drive 
two  stakes,  A  and  B,  (Fig.  68,)  four  or  five  chains  apart. 
At  C,  equidistant  from  them,  set  the  instrument.  Level 
the  plates,  and  bring  the  bubble  in  the  telescope  level,  to 
the  middle  of  its  run ;  then  let  an  assistant  hold  a  graduated 
staff  on  A.  Note  exactly  the  point  in  which  the  line  of 
sight  meets  the  staff:  then  let  the  assistant  remove  the 
staff  to  B,  and  drive  the  stake  B  until  the  telescope  points 


Fig,  68. 


to  the  same  spot  on  the  staff.     The  tops  of  A  and  B  are 
then  level,  whether  the  instrument  is  in  adjustment  or  not. 


106  PLANE  TRIGONOMETRY.  [CHAP.  III. 

"Now  remove  the  instrument  to  G,  and  level  as  before. 
Direct  the  telescope  to  the  staff  on  B,  and  note  the  point 
I  of  intersection.  Let  the  assistant  carry  the  staff  to  A. 
Again  note  the  intersection  K.  If  the  instrument  is 
properly  adjusted,  these  two  points  will  coincide.  If 
they  do  not,  the  line  of  collimation  points  too  high  or  too  low. 

Take  the  difference  between  BI  and  AK  This  differ- 
ence will  be  LK,  the  difference  of  level  as  given  by  the 
instrument  at  G.  Then  say,  As  the  distance  between  the 
stakes  (BA)  is  to  the  distance  from  the  instrument  to  the 
far  stake  (GA),  so  is  the  difference  of  apparent  level  of  the 
stakes  (LK)  to  the  correction  on  the  far  staff  (MK). 

This  correction — either  taken  from  the  height  AK  if  too 
great,  or  added  to  it  if  too  small — will  give  AM,  the  height 
of  a  point  on  the  same  level  as  the  instrument.  Direct  the 
telescope  to  this  point,  and  rectify  the  level,  by  raising  or 
lowering  one  end  by  means  of  the  capstan  screw  until  the 
bubble  is  in  the  middle  of  its  run.  If  the  operation  has 
been  carefully  done,  the  adjustment  is  perfect.  Verify 
again ;  and,  if  needful,  repeat  the  operation. 

193,  Second  Adjustment.     The  zeros  of  the  vernier  and 
of  the  vertical  limb  should  coincide  when  the  telescope  is  level. 

When  the  first  adjustment  is  perfected,  and  the  telescope 
is  still  level,  examine  the  reading  on  the  vertical  limb  care- 
fully: if  the  zeros  coincide,  the  vernier  is  properly  ad- 
justed ;  if  they  do  not,  note  the  error,  and  have  it  marked 
somewhere  on  the  instrument  under  the  plates,  that  it  may 
not  be  forgotten.  It  must  be  applied  to  all  angles  of  eleva- 
tion taken  by  the  instrument. 

If  the  index-arm  is  movable,  as  is  frequently  the  case 
with  transits,  it  should  be  adjusted  before  taking  vertical 
angles. 

194.  When  all  the  preceding  adjustments  have  been  exa- 
mined, and  rectified  if  necessary,  the  instrument  is  ready 
for  work.     It  would  be  well,  however,  to  examine  carefully 
the  reading  of  the  verniers,  to  see  that  they  are  properly 
divided.      However  placed,  no   two  lines  of   the  vernier 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.  107 

except  the  first  and  last  should  coincide  with  divisions  on 
the  arc.  If  two  are  found  to  do  so  in  any  position, 
there  is  an  imperfection  in  the  graduation.  If  the  division 
is  very  fine,  a  number  of  lines  in  the  immediate  neighbor- 
hood of  the  coincident  lines  will  differ  very  slightly  from 
coincidence;  but,  when  carefully  examined  with  a  good 
magnifier,  they  should  recede  gradually. 

Place  the  instrument  where  a  good  view  of  a  fine  point, 
some  eight  or  ten  chains  distant,  can  be  obtained.  Level 
carefully,  direct  the  line  of  sight  to  the  point,  and  note  the 
reading  on  the  horizontal  limb.  Reverse  the  telescope  in 
its  Y's,  or,  if  the  instrument  is  a  transit,  turn  it  over;  turn 
the  vernier-plate  till  the  line  of  sight  passes  again  through 
the  point,  and  note  the  reading.  It  should  differ  by  180° 
from  that  before  obtained.  If  it  does  not,  the  divisions  are 
not  perfect,  or  the  telescope  is  not  over  the  centre  of  the 
plates.  Either  defect  should  condemn  the  instrument,  as  it 
can  be  remedied  only  by  the  maker.  This  verification 
should  be  tried  in  various  positions  of  the  divided  plate. 
If  these  tests,  and  those  formerly  mentioned,  are  found  to 
detect  no  imperfection,  the  instrument  may  be  pronounced 
a  good  one. 

195,  Taking  Angles.  Set  the  instrument  precisely  over 
the  angular  point,  and  level  it,  being  careful  to  have  the 
levelling  screws  pressed  tightly  against  the  plates,  that  the 
instrument  may  be  steady.  Set  the  index  to  zero,  and 
clamp  the  plates,  and,  if  there  be  more  than  one  vernier, 
note  the  minutes  and  seconds  of  the  others.  Loosen  the 
lower  clamp,  and  bring  the  telescope  so  that  the  wires  may 
intersect  on  the  left-hand  object;  clamp,  and  perfect  the  ad- 
justment by  the  tangent  screw.  If  there  is  a  watch-tele- 
scope, set  it  upon  some  well-defined  object, — such  as  a  light- 
ning-rod or  the  corner  of  a  chimney, — and  clamp  it  tightly. 
Loosen  the  vernier-plate,  and  turn  the  telescope  to  the 
other  object,  perfecting  the  adjustment  by  the  tangent 
screw.  Examine  the  watch  telescope,  and,  if  the  instru- 
ment has  shifted,  bring  it  back  by  the  tangent  screw, 
and  readjust  the  telescope  by  moving  the  vernier-plate. 


108 


PLANE  TRIGONOMETRY. 


[CHAP.  III. 


"Now  read  the  arc  by  the  same  index  as  before,  noting  the 
minutes  and  seconds  by  the  other  verniers.  Take  the  mean 
of  the  minutes  and  seconds  of  each  position  for  the  true 
reading.  Then  the  true  reading  in  the  first  position  taken 
from  that  in  the  second  will  give  the  angle  required.  It  is 
convenient  to  have  a  table  prepared,  with  the  requisite 
number  of  columns,  in  which  to  set  down  the  readings  of 
the  different  verniers.  Thus,  suppose  there  were  three 
verniers,  120  degrees  apart :  rule  a  table,  with  six  columns, 
as  below : — 


Occd. 
Sta. 

Obs. 
Sta. 

A 

B 

c 

Mean. 

A 

B 

0°  0'    0" 

0'  30" 

59'  45" 

0°  0'  7J" 

A 

C 

75°  8'  15" 

8'    0" 

8'  30" 

75°  8'  15" 

The  first  column  is  the  occupied  station;  the  second,  the 
observed  station ;  the  next  three  the  readings  of  the  verniers, 
and  the  sixth  the  mean. 

In  the  case  above,  the  angle  BAG  would  be  75°  8'  7J". 
The  instrument  is  supposed  to  read  to  30",  the  15"  being 
taken  when  two  lines  on  the  vernier  appear  equally  near 
coincidence. 

196.  Repetition  of  Angles.  The  following  method  of 
observation  is  sometimes  employed.  Suppose  the  angle 
ABC  is  to  be  measured,  A  being  the  left-hand  object :  direct 
to  A,  and  turn  to  B  as  above  directed.  Clamp  the  vernier- 
plate  and  loosen  below,  and  bring  the  telescope  again  to  A. 
Clarnp  below,  loosen  the  vernier,  and  bring  the  telescope 
again  to  B.  The  index  has  now  traversed  an  arc  measuring 
twice  ABC.  The  operation  may  be  repeated  as  often  as 
desired,  noting  the  number  of  whole  revolutions  the  tele- 
scope has  made.  Then  divide  the  whole  number  of  degrees 
by  the  number  of  repetitions.  The  result  will  be  the 
degrees  of  the  angle  required.  If  there  is  a  watch-telescope, 
it  should  be  set  carefully  before  each  observation.  When 
this  is  done,  and  proper  care  is  taken  to  avoid  deranging 


SEC.  V.]  INSTRUMENTS  AND  FIELD  OPERATIONS.    »  109 

the  instrument,  the  result  may  be  depended  on  as  more 
accurate  than  any  single  reading.  Any  error  in  the  final 
reading,  being  divided  by  the  number  of  observations,  will 
affect  the  result  by  but  a  small  part  of  its  value. 

197.  Verification  of  the  Angles.    When  it  is  possible 
to  do  so,  all  the  angles  of  a  triangle  should  be  measured. 
If  their  sum  does  not  make  180°,  there  must  be  an  error 
somewhere.     Should  the  error  be  considerable,  the  work 
ought  to  be  reviewed.     But  if  it  does  not  exceed  two  or 
three   minutes,   providing    the   instrument   only  reads  to 
•minutes,  it  may  be   distributed  equally  among  the  three 
angles,  should  there  be  no  reason  to  suppose  one  is  more 
accurate  than  another.     But  if  more  observations  have  been 
taken  for  some  angles  than  for  others,  their  determination 
should  be  most  depended  on,  and  a  proportionally  less  part 
of  the  correction  assigned  to  them.     Suppose,  for  example, 
the  angle  A  is  the  mean  of  five  observations,  B  of  three, 
while  at  C  but  one  was  taken,  the  error  being  1'  45" :  we 
would  proceed  thus:— As  J  +  }  +  1  :  i  : :  1'  45"  :  14",  the 
correction  for  A.     In  the  same  manner  the  correction  for  B 
would  be  found  to  be  23",  and  for  C,  V  08". 

198.  Reduction  to  the  Centre.    Where  the  object  that 
has  been  observed  is  a  spire  or  other  portion  of  a  building, 
it  is  impossible  to  set  the  instrument  underneath  the  signal. 
In  such  cases,  the  observed  angle  must  be  reduced  to  what 
it  would  have  been  had  the  station  been  at  the  proper  point. 
Thus,  let  C  (Fig.  69)  be  the  correct  pig.  69. 
station,  and  D  the  occupied  station, 

which   should  be  taken   as   near  as 

possible  to  C.     Take  the  angle  ADB. 

Then  if  A,  C,  D,  and  B  are  all  in  the 

circumference  of  a  circle,  this  will  be 

equal  to  ACB.     The  station  should 

be  assumed  as  near  this  as  possible.     Calculate  BC  and  AC 

from  the  distance  AB  and  the  angles  observed  at  A  and  B. 

Also   measure  DC,  either  directly  or  by  trigonometrical 

methods  to  be  explained  hereafter,  and  take  ADC. 


110  PLAtfE   TRIGONOMETRY.  [CHAP.  III. 

Then,  (Art  139,)  As  CA  :  CD  : :  sin.  ADC  :  sin.  CAD. 
And  as  CB  :  CD  : :  sin.  BDC  :  sin.  CBD. 
Hence,     ACB  =  AEB  —  CAD  =  ADB  +  CBD  —  CAD, 
becomes  known. 

Example.  Let  CA  =  9647  ft. ;  CB  =  8945  ft. ;  ADB  = 
68°  45' ;  DC  =  150  ft;  and  ADC  =  97°  37'. 

As  CA  9647  ft.  A.  C.  6.015608 

:  CD  150ft.  2.176091 

: :  sin.  ADC  97°  37'  9.996151 

:  sin.  CAD  0°  52'  59"  8.187850 

As        CB  8945  ft  A.  C.  6.048420 

:"         CD  150ft  2.176091 

: :  sin.  CDB  166°  22'  9.372373 

:  sin.  CBD  0°  13'  35"  7.596884 

Whence  ACB  =  ADB  +  CBD  —  CAD  =  68°  5'  36". 

199.  Angles  of  Elevation.  In  measuring  angles  of  ele- 
vation, the  instrument  must  first  be  levelled;  the  telescope 
being  then  directed  to  the  object,  the  reading  of  the  vernier 
corrected  for  the  index-error  will  be  the  angle  of  elevation. 


SECTION  VI. 

MISCELLANEOUS  PROBLEMS  TO  ILLUSTRATE  THE  RULES 
OF  PLANE  TRIGONOMETRY, 

Problem  1.  Being  desirous  of  determining  the  height 
of  a  fir-tree  standing  in  my  garden,  I  measured  100  feet 
from  its  base,  the  ground  being  level.  I  then  took  the 
angle  of  elevation  of  the  top,  and  found  it  to  be  47°  50'  30". 
Kequired  the  height,  the  theodolite  being  5  feet  from  the 
ground. 


SEC.  VL] 


MISCELLANEOUS  PROBLEMS. 


Ill 


Fig.  70. 


Solution. 

Make  AB  (Fig.  70)  equal  to  100  feet; 
draw  AD  and  BC  perpendicular  to  AB, 
making  the  former  five  feet  from  the  same 
scale.  Draw  DE  parallel  to  AB,  and 
make  EDO  =  47°  50',  the  given  angle. 
Then  will  CB  be  the  height  of  the  tree. 

Calculation. 

As  rad.  :  tan.  EDO  : :  DE  :  EC  =  110.45  feet; 
whence  BC  =  110.45  +  5  =  115.45. 


Problem  2.  One  corner  C  (Fig.  71)  of 
a  tract  of  land  being  inaccessible,  to  de- 
termine the  distances  from  the  adjacent 
corners  A  and  B,  I  measured  AB  =  9.57 
chains.  At  A,  the  angle  BAG  was  52°  19' 
15",  and  at  B,  the  angle  ABC  was  63°  19' 
45".  Required  the  distances  AC  and  BC.  B 


Calculation. 

As  sin.  ACB  (64°21')  :  sin.  A  (52°  19'  15")  : :  AB  (957)  : 
BC  =  840.2  links.  As  sin.  ACB  (64°  21')  :  sin.  B  (63°  19' 
45")  : :  AB  :  AC  =  948.7  links. 


Problem  3.  In  measuring  the  sides 
of  a  tract  of  land,  one  side  AB  (Fig. 
72)  was  found  to  pass  through  a  swamp, 
so  that  it  could  not  be  chained.  I  there- 
fore selected  two  stations,  C  and  D,  on 
fast  land,  and  took  the  distances  and  angles  as  follows, — 
viz.:  AC  =  37.56  chains;  CD  =  50.25  chains;  BAC  = 
65°  27'  30";  ACD  =  123°  46'  20";  CDB  =  107°  29'  15": 
the  corner  B  being  inaccessible,  the  distance  BD  could  not 
be  measured.  Required  AB.  The  angle  CDA  could  not  be 
taken,  owing  to  obstructions. 


112  PLANE   TRIGONOMETRY.  [CHAP.  III. 

Solution. 

Join  AD.     Then,  from  the  triangle  ACD,  we  have,  (Art. 
140,) 

As  CD  +  CA  (87.81) :  CD  -  CA  (12.69) : :  tan. 
CAD- 


CAD  +  CDA 

~~2 


(28°  6'  50")  :  tan. 


,  54/,. 


whence  CAD  =  28°  6'  50"  +  4°  24'  54"  =  32°  31'  44", 
and  CDA  =  28°  6'  50"  -  4°  24'  54"  =  23°  41'  56"  ; 
then,  sin.  CDA  :  sin.  ACD  :  :  AC  :  AD  =  TT.68. 

Now,  in  ADB  we  have  AD  =  77.68,  the  angle  DAB  =  CAB 
—  CAD  =  32°  55'  46",  and  the  angle  ADB  =  BDC  -  ADC 
=  83°  47'  19",  to  find  AB;  thus, 

As  sin.  B  :  sin.  ADB  :  :  AD  :  AB  =  86.455  chains. 


Fig.  73. 


Problem  4.  To  determine  the  position  of  a  point  D  on 
an  island,  I  ascertained  the  distances  of  three  objects  on  the 
main  land  as  follows:—  AB  =  248.75  chains,  BC  =  213.25 
chains,  and  AC  =  325.96  chains.  At  D  the  angle  ADB  was 
found  to  be  29°  15',  and  BDC  20°  29'  30".  Required  the 
distance  of  D  from  each  of  the  objects. 

Construction. 

With  the  given  distances  construct 
the  triangle  ABC.  At  C  and  A  make 
the  angles  ACE  =  29°  15',  and  CAE 
=  20°  29'  30".  About  AEC  describe 
the  circle  ACD.  Join  EB,  and  pro- 
duce it  to  D,  which  will  be  the  point 
required. 

For  (21.3)  ADB  =  ACE  =  29°  15', 
and  CDB  =  CAE  =  20°  29'  30". 

Calculation. 

1.  In  ABC  we  have  the  three  sides  to  find  the  angle  BAC 
=  40°  51'  30". 

2.  In  CAE  we  have  the  angles  and  side  AC  to  find  the 
side  AE  =  208.705. 

3.  In  BAE  we  have  BA,  AE,  and  the  included  angle 
BAE,  to  find  ABE  =  50°  55'  48",  AEB  =  67°  43'  12". 


SEC.  VI.]  MISCELLANEOUS   PROBLEMS.  113 

4.  In  ABD  we  have  the  angles  and  side  AB,  to  find  AD 
=  395.24  and  BD  =  188.0T. 

5.  In  A  CD  we  have  the  angles  and  sides  AC,  to  find  CD 
=  379. 

Problem  5. — Wishing  to  obtain  the  distance  between  two 
trees,  C  and  D,  situated  on  Fig.  74. 

the  side  of  a  hill,  and  not 
being  able  to  find  level 
ground  for  a  base,  I  select- 
ed a  gradual  slope,  on  which 
I  measured  the  distance  AB 
(Fig.  74)  400  yards.  I  then 
took  the  horizontal  and  ver- 
tical angles  as  follow: — At 
A,  the  angle  BAD  was  101° 
47'  15",  BAC  39°  25'  45".  The  elevation  of  B  was  5°  32' 
45",  of  C,  8°  19'  30",  and  of  D,  12°  29'.  At  B,  the 
angle  ABD  was  59°  13' 15",  and  ABC  125°  36'  45". 

Required  the  distance  CD,  and  the  elevations  of  C  and 
D  above  A. 

Conceive  a  horizontal  plane  to  pass  through  A,  meeting 
vertical  lines  through  B,  C,  and  D  in  the  points  E,  F,  and  G. 
Then,  since  the  angular  distances  are  measured  horizontally, 
we  have  the  following  angles  given, — viz. :  EAG  ==  101°  47' 
15",  EAF  =  39°  25'  45",  AEG  =  59°  13'  15",  and  AEF  = 
125°  36'  45". 

Calculation. 

1.  To  find  AE,  we  have  r  :  cos.  BAE  (5°  32'  45")  : :  AB 
(400) :  AE  =  398.13. 

2.  To  find  AG.    As  sin.  AGE  :  sin.  AEG  : :  AE  :  AG  = 
1051.07,  log.  3.021631. 

3.  To  find  AF.    As  sin.  AFE  :  sin.  AEF  : :  AE  :  AF  = 
1253.96,  log.  3.098284. 

4.  TofindFG,(Art.l41.)AsAG:AF::r:tan.z  =  50°l'49". 
And,  as  rad.  :  tan.  (x  -  45°)  : :  tan.  |  (AGF  +  AFG)  :  tan. 

J  (AGF  -  AFG)  =  8°  16'  34"; 

then      AGF  =58°  49' 15" +  8°  16' 34"  =  67°    5' 49", 

and       AFG  =  58°  49'  15"  -  8°  16'  34"  =  50°  32'  41". 


114  PLANE  TRIGONOMETRY.  [CHAP.  III. 

Then,     as  sin.  AGF  :  sin.  FAG  : :  AF  :  GF  =  1205.9. 

5.  To  find  GD  and  OF.     As  r  :  tan.  GAD  : :  AG  :  GD  = 
232.69  =  Elevation  of  D. 

And  as  r :  tan.  OAF  : :  AF :  FC  =  183.49  =  Elevation  of  C. 

6.  To  find  CD.      CD  =  ^  CH2  +  HD3  =  1206.9  =  Dis- 
tance of  CD. 

Problem  6. — Being  desirous  to  determine  the  height  of 
a  tower  standing  on  the  summit  of  a  hill,  I  measured  75 
yards  from  its  base  down  the  declivity,  which  was  a  regular 
slope.  I  then  took  the  elevation  of  the  top,  49°  37'  45",  and 
of  the  bottom,  8°  19',  the  height  of  the  instrument  being  5 
feet.  What  was  the  height  of  the  tower  ?  Ans.  76.44  yds. 

Problem  7. — To  determine  the  height  of  a  tree  in  an 
inaccessible  situation,  I  took  a  station,  and  found  the  ele- 
vation of  the  top  to  be  38°  45'  15" ;  then,  measuring  back 
100  feet,  the  elevation  was  found  to  be  24°  18'.  Required 
the  altitude  of  the  tree  and  its  distance  from  the  first  sta- 
tion, the  instrument  being  4  feet  9  inches  high. 

Ans.  Height,  107.95  feet;  distance.  128.57  feet. 

Problem  8. — To  determine  the  distance  of  two  objects 
A  and  B,  I  took  two  stations  C  and  D,  distant  35.75  chains, 
from  which  both  could  be  seen.  At  C,  the  angle  ACD  was 
found  to  be  103°  47',  and  BCD  45°  29'  30" ;  at  D,  the  angle 
BDC  was  110°  23'  30",  and  ADC  60°  21'  15".  Required  the 
distance  AB.  Ans.  99.236  ch. 

Problem  9— The  side  AB  (Fig.  75)  of  a  tract  of  land 
being  inaccessible,  and  not  being  able  to  find  two  stations 
from  which  both  ends  were  visible, 
I  measured  two  lines,  CD,  7.75  ch., 
and  DE,  7.92  ch.,  and  took  the  angles 
as  follow :  At  C,  the  angle  ACD  was 
68°  15'.  At  D,  CDA  was  50°  27', 
ADB  112°  46',  andBDE  43°  30'. 
E,  DEB  was  75°  10'.  What  was  the  length  of  AB  ? 

Ans.   14.10  ch. 


SBC.  VI.] 


MISCELLANEOUS   PROBLEMS. 


115 


Problem  10, — To  determine  the  position  of  a  point  D, 
situated  on  an  island,  I  took  the  angles  to  three  objects, 
A,  B,  and  C,  situated  on  the  shore,  and  found  them  to  be 
ADB,  19°  14'  30",  CDB,  24°  19'.  I  subsequently  deter- 
mined the  distances  AB  =  4596  yards,  AC  =  5916  yards, 
and  BC  =  4153  yards.  Required  the  distance  of  D  from 
each  of  the  objects,  it  being  nearest  to  B. 

Ans.  AD  =  828T.2  yards ;  BD  =  412T.7  yards ;  CD  = 
7550.8  yards. 

Problem  11. — To  determine  the  height  of  a  mountain 
rising  abruptly  from  the  water  of  a  lake,  I  selected  a  station 
C  on  the  slope  of  the  hill  rising  from  the  opposite  shore,  and 
took  the  angle  of  elevation  of  the  summit,  47°  22'  15",  and 
depression  of  the  water's  edge  at  the  base  of  the  mountain 
in  the  vertical  plane  through  the  summit,  12°  30'.  Then 
measuring  up  the  slope,  directly  from  the  rock,  a  distance  of 
800  yards,  to  a  station  D,  the  elevation  of  the  summit  was 
25°  33'  30",  the  depression  of  the  water's  edge,  18°  15',  and 
of  the  top  of  a  staff  left  at  C  to  mark  the  height  of  the 
instrument,  24°  15'.  Required  the  height  of  the  mountain. 

Ans.  Height,  1390.7  yds. 

Problem  12. — To  determine  the  heights  and  distance  of 
two  trees  C  and  D,  standing  on  a  hill  side,  I  measured  on  level 
ground  a  base  line  AB  252.28  feet 
long,  and  took  the  following  angles : 
At  A,  the  angle  of  position  of  C  from 
B  was  =  82°  54'  30",  and  of  D  from 
B  =  89°  24' ;  the  elevation  of  the 
base  of  C  =  3°  45' ;  of  top  of  do.  = 
9°  25' ;  of  the  base  of  D  =  3°  54' ; 
of  top  of  do.  =  10°  29'  30".  At  B, 
the  angle  of  position  of  D  from  C 
was  =  6°  14'  30" ;  and  of  A  from  C 
=  80°  51'  30",  and  for  verification 
the  elevations  at  B  were  of  base  of  C  =  3°  44',  of  top  of 
do.  =  9°  22'  15" ;  of  base  of  D  =  3°  46',  and  of  top  of  do.  = 


Fig.  76. 


116 


PLANE   TRIGONOMETRY. 


[CttAP.  ID. 


Fig.  77. 


10°  7'  30".     Eequired  the  heights  of  the  trees,  and  the  dis- 
tance between  their  bases. 

Ans.  Height  of  C  =  89.37  ft. ;  of  D  =  103.37  ft. ;  dis- 
tance, 100.7  ft.  With  the  angles  of  verification ;  height  of 
C  =  103.29  ft.;  of  D  =  89.36  ft. 

Problem  13.— One  side  EF  (Fig.  77)  of  a  tract  of  land 
being  inaccessible,  and  there  being  no  station  from  which 
the  two  ends  could  be  seen,  I  selected  four 
stations,  A,  B,  C,  D ;  A  and  D  being  in  the 
adjoining  sides,  and  B  and  C  between 
them.  The  following  measurements  were 
then  taken,— viz. :  AB  =  7.37  ch. ;  BC  = 
8.95  ch.,  and  CD  =  9.33  ch. ;  at  A,  the  angle 
EAB  was  64°  37';  at  B,  ABE  was  72°  43', 
and  EBC  149°  32' ;  at  C,  BCF  was  139° 
47',  and  FCD  69°  38' ;  and  at  D,  CDF  was 
82°  35'.  Required  AE,  EF,  FD,  and  the 
angles  AEF  and  EFD. 


Ans.  EF  =  33.50;  AE  =  10.38;  DF  =  18.77; 
AEF  =  86°  39' ;  EFD  =  54°  29'. 

Problem  14. — Being  desirous  of  finding  the  elevation  and 
distance  of  an  elevated  peak  C  (Fig. 
78)  of  a  mountain  rising  abruptly 
from  the  shore  of  a  river,  and  not 
being  able  to  find  a  level  place  for 
a  base  line,  or  a  regular  slope  as- 
cending in  a  line  from  the  point  to 
be  measured,  I  selected  two  stations, 
the  one  A  nearly  opposite  the  base  D 
of  a  rock  jutting  into  the  water,  and 
which  was  so  situated  that  A,  C,  and 
D  were  in  the  same  vertical  plane, 
and  the  other  station  B  farther  up  the  stream,  the  slope 
between  them  being  regular.  I  then  took  the  following 


Fig  78. 


SEC.  VI.]  MISCELLANEOUS   PROBLEMS.  117 

measurements, — viz. :  AB,  850  yards.  At  A,  the  angle  of 
position  of  B  and  C  was  87°  49';  elevation  of  C,  35°  27'; 
depression  of  D,  3°  25'  45";  elevation  of  top  of  a  staff  at 
B  of  same  height  as  the  instrument,  3°  14'  30".  At  B,  the 
angle  of  position  of  A  and  D  was  47°  39',  and  of  A  and  C, 
70°  43'  30".  Depression  of  A,  3°  14'  30" ;  of  D,  4°  48'  30" ; 
elevation  of  C,  33°  6'.  Required  the  horizontal  distance  of 
C  and  D  from  A  and  B,  and  the  elevation  of  A,  B,  and  C 
above  the  water. 

Ans.  Horizontal  distance  of  C  from  A,  2189.8  yds. ;  from 
B,  2318.1  yds. ;  of  D  from  A,  894.3  yds. ;  from  B,  1209.2yds. 
Elevation  of  C,  1612.7  yds. ;  of  A,  53.6  yds. ;  and  of  B  101.7 
yds. 


CHAPTER  IV. 

CHAIN  SURVEYING. 


SECTION  I. 
DEFINITIONS, 

V 

200.  Definition.     LAND  SURVEYING  is  the  art  of  mea- 
suring the  dimensions  of  a  tract  of  land,  so  as  to  furnish 
data  for  calculating  the  content  and  determining  the  area. 

201.  The  position  of  the  angular  points  of  a  tract  may 
be  determined  either  by  measuring  the  lines  of  the  survey, 
the  diagonals,  offsets,  &c.,  or  by  linear  measures  in  connection 
with  angular  distances.     These  different  methods  of  fixing 
the  points  give  rise  to  different  modes  of  surveying, — the 
first  of  which,  as  it  is  performed  principally  by  the  chain, 
may  be  called  chain  surveying. 

202.  Advantages.    As  the  chain,  or  some  substitute,  such 
as  a  tape-line  or  a  cord,  is  readily  procured  by  every  one, 
surveying  by  this  method  may  be  performed  where  the 
more  expensive  instruments   cannot  readily  be  procured. 
To  every  farmer  it  may  be  important  to  know  the  content 
of  a  particular  field,  or  of  several  fields,  that  he  may  divide 
them  properly,  or  that  he  may  know  the  value  of  crops 
which  he  is  about  to  buy  or  to  sell ;   or  for  various  other 
purposes  that  need  not  be  mentioned.     He  should,  there- 
fore, not  be  under  the  necessity  of  calling  in  a  professional 
man  to  do  for  him  what  he  himself,  with  a  pair  of  carriage 
lines,  can  do,  if  not  as  well,  yet  fully  well  enough  for  all 
practical  purposes. 

118 


SEC.  II.]  FIELD  OPERATIONS.  119 

In  order  that  this  very  simple  method  may  be  fully 
understood,  we  shall  treat  of  it  somewhat  at  length.  It 
must  not  be  inferred  from  this  that  it  is  recommended  in 
preference  to  the  other  methods  to  be  explained  here- 
after, but  .only  as  a  substitute  to  be  used,  when,  from  the 
circumstances  of  the  case,  these  are  inapplicable  or  incon- 
venient. 

203.  Area  Horizontal.  It  must  be  remembered  that, 
in  land  surveying,  it  is  the  horizontal  area  that  is  required, 
and  not  the  actual  surface  of  the  ground.  Every  measure- 
ment must,  therefore,  be  made  horizontally,  as  explained 
in  Art.  149,  et  seq.,  and,  where  angles  are  taken,  they  must 
be  horizontal  angles. 

As  the  method  of  chaining  has  been  fully  explained  in 
the  articles  above  referred  to,  it  will  be  unnecessary  to 
repeat  the  directions  here.  There  are,  however,  certain 
preliminary  operations  to  be  performed,  which  will  form 
the  subject  of  the  next  section. 


SECTION  II. 

FIELD  OPERATIONS, 

A.— TO  RANGE   OUT   LINES,   AND   TO   INTERPOLATE 

POINTS. 

204.  Ranging  out  Lines.  THIS  requires  three  persons, 
each  of  whom  should  be  provided  with  a  rod  some  ten  or 
twelve  feet  long,  one  end  being  pointed  with  iron,  that  it 
may  be  thrust  in  the  ground.  lie  should  also  have  a 
plumb-line,  that  he  may  set  his  rod  upright.  The  first, 


120  CHAIN  SURVEYING.  [CHAP.  IV. 

whom  we  shall  call  A,  takes  his  station  at  the  point  of  be- 
ginning. Looking  in  the  direction  of  the  line,  he  places  B 
in  the  proper  direction,  signalling  him  to  the  right  or  left 
as  may  be  required.  "When  the  position  is  determined,  B 
sets  his  rod  firmly  in  the  ground.  C  then  goes,  forward, 
and  looking  back,  by  ranging  with  the  rods  of  B  and  A,  he 
puts  his  rod  in  line.  A  then  comes  forward,  and,  going 
ahead  of  C,  puts  himself  in  line,  by  ranging  with  C  and  B. 
They  thus  continue,  the  hindmost  always  coming  forward, 
until  the  other  end  of  the  line  is  reached.  At  the  point  at 
which  each  rod  was  erected  a  stake  should  be  driven  for 
future  reference. 

Lines  may  be  prolonged  in  the  same  manner  to  any 
extent  that  may  be  desired. 

If  the  operation  is  carefully  done,  the  rods  being  set 
plumb,  the  line  will  vary  very  slightly,  if  at  all,  from  a 
straight  line,  even  when  extended  several  miles. 

205.  To  interpolate  points  in  a  line.     The  men  in 

chaining  should  keep  themselves  exactly  in  line.  This 
may  readily  be  done  by  a  careful  follower/ when  the  end 
of  the  line  can  be  seen.  If,  however,  one  end  is  not  visi- 
ble from  the  other,  and  from  every  point  in  the  line,  there 
will  be  nothing  by  which  the  follower  can  range  his  leader, 
unless  there  are  staves  set  up  for  that  purpose,  at  points 
along  the  line.  The  fixing  of  such  points  is  called  inter- 
polation. 

206.  On  level  ground.     If,  for  any  purpose,  such  points 
were  needed  in  a  line  on  level  open  ground,  a  person, 
stationing  himself  at  one  end,  can  signal  another  into  the 
proper  position.     As  many  points  as  are  wanted  can  thus 
be  determined. 

207.  Over  a  hill.     If  a  hill  intervenes,  from  the  top  of 
which  both  points  may  be  seen,  let  two  persons,  provided 
with   rods,   put   themselves   as   near  in   line   as   possible. 
Then,  by  alternately  signalling  to  each  other,  their  proper 


SEC.  II.]  FIELD  OPERATIONS.  121 

places  can  be  found.  Thus,  let  XY  (Fig.  79)  be  the  Kg.  w. 
line  to  be  interpolated.  A  will  take  his  station  in 
the  supposed  position  of  the  line,  and  signal  B 
until  he  ranges  with  X.  B  then  places  A  in  line 
with  Y  at  .0 ;  A  again  signals  B  to  D,  in  line  with 
X ;  and  so  they  proceed  till  they  are  both  in  the 
line  XY. 

208.  If  an  assistant  is  not  at  hand,  or  if  but 
one  point  can  be  found  from  which  both  ends  of 
the  line  can  be  seen,  one  person  can  put  himself 
in  line  by  having  a  rule  with  a  sight  at  each  end ; 
wires,  set  upright,  will  do  very  well :  lay  this  on 
some  support,  and  then  go  to  each  end  in  turn, 
sighting  to  the  end  of  the  line ;  he  can  thus  deter- 
mine whether  it  is  the  proper  position,  and  alter  it  until  he 
finds  himself  rightly  placed. 

209.  By  a  Random  Line.    "When  the  ends  cannot  be 
seen  from  each  other,  nor  from  any  intermediate  point,  it  is 
necessary  to  run  a  random  line.     This  is  done  as  directed 
in  Art.  204,  following  a  course  as  near  that  of  the  line  to 
be  interpolated  as  possible. 

When  the  foremost  person  has  come  opposite  the  end  of 
the  line,  measure  the  whole  length,  noting  the  distance  to 
each  stake,  (the  stakes,  for  convenience,  being  set  as  nearly 
as  possible  at  equal  distances ;)  also  measure  the  distance 
by  which  the  end  of  the  line  is  missed,  then  say : — 

As  the  whole  distance  is  to  the  distance  to  any  stake,  so 
is  the  whole  deviation  to  the  correction  for  that  stake. 
Measure  the  distance  thus  determined,  in  the  proper  di 
rection,  and  set  the  stake,  or  a  stone,  accordingly. 


122  CHAIN  SURVEYING.  [CHAP.  IV. 

Thus,  let  AB  (Fig.  80)  be  the  line  to  be  inter-      *fe.  so. 
polated.     Run  the  random  line  AC,  setting  stakes 
at  D,  E,  F,  &c.      Measure  CB  and  the  distance 
from  A  to  J>,  E,  F,  and  C. 

Suppose  AC  measures  27.56  chains,  AD  10 
chains,  AE  15  chains,  AF  20  chains,  and  BC  = 
1.57  chains. 

Then,  27.56  :  10  : :  1.57  :  .57,  the  correction  for  D. 
Similarly,      Ee  =  .85,  and  F/=  1.14  chains. 

Set  off  Dd,  Ee,  and  F/,  the  calculated  distances; 
set  stakes  at  d,  e,  and  /,  and  range  out  the  line 
anew. 

Instead  of  working  out  each  proportion,  it  is 
more  concise  to  divide  the  deviation  by  the  num- 
ber of  chains  in  the  measured  length :  this  will  give  the 
correction  for  one  chain.      This  correction,  being  multi- 
plied by  the  distance  to  each  stake,  will  give  the  correction 
for  that  stake. 

Thus,  in  the  above  example, 

1.57 
— —  =   .057,  the  correction  for  1  chain. 

10  X  .057  =  .57,  the  correction  for  D ; 
15  x  .057  =  .85,  the  correction  for  E ; 
20  x  .057  =  1.14,  the  correction  for  F. 

210.  Across  a  valley.  When  the  line  runs  across  a 
valley,  let  two  points  A  and  B  be  determined  on  opposite 
sides  of  the  valley,  from  which  the  intervening  ground  can 
be  seen.  Then  let  one  person  take  his  station  at  A,  and, 
holding  a  plumb-line  over  the  stake,  let  him  sight  to  B :  he 
can  then  direct  his  assistant  into  the  proper  position,  and 
thus  fix  as  many  points  as  are  desirable. 

NOTE. — These  operations  are  all  done  more  accurately  and  rapidly  by  means 
of  the  transit  or  theodolite. 


SEC.  II.]  FIELD  OPEKATIONS.  123 

211,  To  determine  the  point  of  intersection  of   two  visual 
lines. 

This  is  most  readily  done  by  three  persons,  two  of 
whom  take  their  stations  in  the  lines,  at  some  distance 
from  the  point  of  intersection,  and,  looking  along  their 
lines  respectively,  signal  the  third  until  he  ranges  in  both 
lines.  A  stake  may  then  be  driven  at  the  point  of  inter- 
section. 

This  operation  may  readily  be  performed  by  two  persons. 
First,  let  them  run  out  one  of  the  lines,  and  stretch  a  cord 
or  the  chain  across  the  course  of  the  other.  One  of  them 
then  taking  his  station  in  the  second  line  can  signal  the 
other  to  his  proper  position. 

212.  To  run  a  line  towards  an  invisible  intersection. 

Through?  (Fig.  81)  Fig.81. 

run  the  line  AC,  in- 
tersecting the  given 
lines  in  A  and  C. 
Then  through  any 
point  B  in  AB  set  out 
BD  parallel  to  AC  by 
one  of  the  modes  to  be  pointed  out.  (See  Arts.  227-229.) 
Divide  BD  in  F,  so  that  BF  :  FD  : :  AP  :  PC;  that  is, 

make  BF  = -- — .     Then  PF  will  be  the  required  line. 

AC* 


B.— PERPENDICULARS. 

Problem  1. —  To  draw  a  perpendicular  to  a  given  line  from  a 
given  point  in  it. 

213.  (a.)  When  the  Point  is  accessible.  This  may  be 
done  on  the  ground  by  the  methods  described  in  Arts.  88, 
89,  and  90,  using  the  chain  for  a  pair  of  compasses  to  sweep 
the  circles,  or  by  the  following  methods : — 


124 


CHAIN  SURVEYING. 


[CHAP.  IV. 


214.    First    Method.    Let  AB  Fig.  82. 

(Fig.  82)  be  the  line  and  C  the 
point  at  which  the  perpendicular 
is  to  be  erected.  First,  lay  off 
CD,  60  links;  then,  fixing  one 
end  of  the  chain  at  D,  sweep  an 
arc  of  a  circle  at  E,  using  the 

whole    chain  (100  links)   for  a     j- 

radius.    Next,  fix  one  end  at  C, 

and,  with  80  links  for  a  radius,  sweep  an  arc  cutting  the 

former  in  E.     CE  will  be  perpendicular  to  AB. 

Any  other  distances,  in  the  same  ratio  as  the  above,  will 
answer.  Thus,  DC  might  be  30,  CE  40,  and  DE  50. 
"With  these  numbers  no  circles  need  be  struck.  Lay  off 
DC  =  30  links;  fix  the  end  of  the  chain  at  D,  and  the  end 
of  the  ninetieth  link  at  C :  then,  taking  the  end  of  the 
fiftieth  link,  stretch  both  parts  of  the  chain  equally  tight, 
and  set  a  stake  at  the  point  of  intersection. 

These  numbers  are  very  convenient  when  short  perpen- 
diculars are  required ;  but  when  the  line  is  run  to  some  dis- 
tance the  greater  lengths  are  preferable. 


215.  Second  Method.  Make  AC 
(Fig.  83)  a  chain.  With  the  whole 
length  of  the  chain  sweep  two  arcs 
cutting  in  D ;  range  out  AD,  making 
DE  =  AD :  then  CE  will  be  the  per- 
pendicular required. 

For,  ADC  being  equilateral,  A= 
60°,  and  A  and  ACD  =  120°;  whence 
DCE  and  DEC  =  60°.  But  DE  =  DC : 
therefore  DCE  =  30°,  and  ACE  =  90°, 


Fig.  83. 


SEC.  II.] 


FIELD  OPERATIONS. 


125 


216.  (o.)  When  the  Point  is  inaccessible. 

Erect  a  perpendicular  at 
some  other  point  D  (Fig.  84)  of 
the  line.  Through  F,  a  point 
in  this  perpendicular,  draw 
FH  parallel  to  AB,  (Art.  227.) 
Take  FE  =  FD :  range  out  EC, 
intersecting  FH  in  Q-.  Make 
GH  equal  FG:  then  CHI  will 
be  the  perpendicular  required. 

FE  need  not  be  taken  equal  to  DF.   If  unequal,  GH  will 
be  determined  by  the  proportion  EF  :  FD  : :  FG  :  GH. 

(c.)  If  the  line  is  inaccessible,  trigonometrical  methods 
must  be  employed. 

Problem  2.     To  let  fall  a  perpendicular  to  a  line  from  a 

point  without  it. 

(a.)   When  the  point  and  line  are  both  accessible. 

217.  The  methods  in  Arts.  91, 92, 
93,  may  be  adopted  in  this  case; 
or  in  AB  (Fig.  85)  take  any  point 
D,  and  measure  CD.   Make  DE  = 
DC,  and  measure  CE. 

FP2 

Then  take  EF  =  -• ,  and  F    ~X~  D       r      i.; 

2.ED 

will  be  the  foot  of  the  perpendicular. 

Describe  the  semicircle  ECA.     Then,  if  CF  is  perpen- 
dicular to  AB,  EC  is  a  mean  proportional  between  AE 

EC2        EC2 
and  EF,  whence  EF  =  —  = 


126 


CHAIN   SURVEYING. 


[CHAP.  IV. 


Fig.  86. 


(b.)  If  the  point  is  remote  or  inaccessible. 

218.  First  Method.—  In  AB 
(Fig.  86)  take  any  convenient 
points  A  and  D;  erect  the 
perpendicular  FDE,  making 
FD  =  DE;  range  out  AE, 
and  EC  cutting  AB  in  H,  and 
FH  intersecting  AE  in  G: 
then  GBC  will  be  perpen- 
dicular to  AB. 


For,  by  construction,  the  triangles  ADE  and  ADF,  as  also  FDH  and  EDH,  are 
equal  in  all  respects.  Hence,  AFG  and  AEG,  having  two  angles  and  the  included 
side  of  one  equal  to  two  angles  and  the  included  side  of  the  other,  are  equal 
in  all  respects ;  therefore  AG  =  AC.  Finally,  ABC  and  ABG  have  two  sides 
and  their  included  angles  respectively  equal,  whence  B  is  a  right  angle. 


219.  Second  Method.— Select 
any  two  convenient  stations  E 
and  F  (Fig.  87)  from  which  C 
may  he  seen,  and  range  out  FC 
and  EC.  To  these  draw  the 
perpendiculars  EG  and  FH  cut- 
ting in  I :  then  CLD  will  he  the 
perpendicular  required. 


For  the  perpendiculars  to  the  three  sides  of  a  triangle  from  the  opposite 
angles  intersect  in  the  same  point. 


(c.)  If  the  line  be  inaccessible. 

220.  From  the  given  point 
C  towards  two  visible  points  A 
and  B  (Fig.  88)  of  the  given 
line  range  out  CA  and  CB, 
and  by  one  of  the  preceding 
methods  draw  the  perpen- 
dicular EA  and  BD  inter- 
secting in  F :  CF  will  be  the 
perpendicular  required. 


Fig.  88. 


221.  The  preceding  methods  will  apply  in  all  the  cases 


SEC.  II. ]  FIELD  OPERATIONS.  127 

enumerated.  They  are,  however,  only  to  be  considered  as 
substitutes  for  the  neater  and  more  accurate  methods  by 
the  use  of  the  theodolite  or  transit.  Measurements  such 
as  those  directed  above,  when  they  are  intended  to  de- 
termine the  direction  of  an  important  line,  require  to  be 
made  with  scrupulous  accuracy ;  for  every  deviation  will  be 
magnified  as  we  proceed.  An  error  of  two  or  three  inches, 
which  would  be  a  matter  of  but  little  importance  in  a  line 
of  a  chain  long,  would  cause  a  deviation  of  from  twelve  to 
twenty  feet  if  the  line  were  prolonged  to  a  mile. 

In  the  absence  of  a  transit  or  theodolite,  the  following 
simple  instruments,  either  of  which  can  be  constructed  by 
any  one  having  a  moderate  degree  of  facility  in  the  use  of 
tools,  will  enable  the  surveyor  to  lay  out  perpendiculars 
with  readiness  and  considerable  accuracy. 

222.  The  Surveyor's  Cross.    This  consists  of  a  block 
of  wood  four  or  five  inches  in  diameter,  with  two  saw-cuts 
across  its  centre  precisely  at  right  angles.     An  auger  hole 
should  be  made  at  the  bottom  of  each  saw-cut,  to  afford  a 
larger  field  of  view.     The  block  is  fastened  to  the  top  of  a 
staff  about  eight  or  ten  inches  long.     It  should  turn  freely 
but  firmly  on  the  head  of  the  staff. 

Instead  of  saw-cuts,  four  wires  may  be  set  upright  at  the  ex- 
tremities of  perpendicular  diameters ;  but,  as  these  are  likely 
to  be  deranged,  the  other  form  is  better. 

223.  To  erect  a  perpendicular  with  the  cross,  set  it  up  at 
the  point  at  which  the  perpendicular  is  to  be  drawn,  and 
turn  it  round  till  one  of  the  cuts  ranges  with  the  given  line; 
then,  looking  through  the  other  cut,  the  surveyor  can  direct 
his  assistant  to  set  a  stake  in  the  required  perpendicular. 

If  the  point  is  out  of  the  line,  take  a  station  as  near  as 
the  eye  can  judge  to  the  position  of  the  foot  of  the  per- 
pendicular, and,  having  set  the  cross  so  that  one  cut  may 
range  with  the  given  line,  look  through  the  other,  and  see 
how  far  the  line  of  sight  misses  the  given  point.  Move  the 
cross  that  distance  and  test  it  again.  A  few  trials  will  de- 
termine the  proper  position. 


128 


CHAIN  SURVEYING. 


[CHAP.  IV. 


224.  To  verify  the  Accuracy  of  the  Cross.     Place  it 
at  a  given  station:  range  with  one  of  the  cuts  to  a  well- 
defined  object,  and  place  a  stake  in  the  perpendicular;  then 
turn  the  cross  one-quarter  round,  and  if  the  stake  is  in  the 
perpendicular,  the  cross  is  correct,  but  if  not,  the  instru- 
ment is  in  error  by  half  the  observed  deviation. 

This  will  be  apparent  by 
reference  to  Fig.  89.  If  the 
angle  A  CD  is  acute,  the 
stake  will  be  placed  to  the 
left  of  the  true  position,  as 
at  F.  By  turning  the  block 
one-fourth  round,  the  acute 
angle  will  be  found  at  BCE, 
and  the  stake  will  be  posited 
at  G-,  as  far  to  the  right  as  it  was  before  to  the  left. 

225.  The  Optical  Square.    The  optical  square  is  a  much 
more   convenient  instrument  for  drawing  perpendiculars 
than  the  cross.     It  consists  of  a  circular  box,  having  a  fine 
vertical  slit  cut  in  one  side,  and  directly  opposite  a  circular 
or  oval  opening  with  a  vertical  line,  such  as  a  horsehair 
stretched  across  it.     The  box  contains  a  piece  of  looking- 
glass  set  across  it,  so  as  to  make  an  angle  of  45°  with  the 
line  of  sight.      From  the  upper  half  of  this  glass  the  sil- 
vering must  be  removed.     Half-way  between  the  two  open- 
ings mentioned  is  another,  to  allow  the  rays  coming  from 
an  object  in  the  perpendicular  to  fall  on  the  mirror  and  be 
reflected  to  the  eye. 


SEC.  II.  ] 


FIELD   OPERATIONS. 


129 


Fig.  90. 


Fig.  90  represents  a 
plan  of  this  instru- 
ment. ABC  is  a  sec- 
tion of  the  box,  A  the 
slit  at  which  the  eye  is 
placed,  B  the  opening 
in  the  line  of  sight, 
C  the  opening  for  the 
perpendicular,  and  DE 
the  looking-glass. 

The  surveyor  holds 
the  box  in  his  hand, 
and,  looking  at  the  other  end  of  the  line,  through  the  open- 
ings A  and  B,  directs  his  assistant,  who  is  seen  by  reflec- 
tion through  C,  to  place  his  rod  in  such  a  position  that  its 
image  shall  coincide  with  the  hair  across  the  opening  B. 
HG  is  then  perpendicular  to  AF. 

To  find  the  point  in  which  the  perpendicular  from  a  dis- 
tant point  will  intersect  AF,  walk  along  the  line,  keeping 
the  line  of  sight  AB  directed  to  the  end  of  the  line.  When 
the  image  of  a  pole  standing  at  the  point  from  which  the 
perpendicular  is  to  be  drawn  appears  at  H,  the  proper  posi- 
tion has  been  attained. 

226.  To  test  the  Accuracy  of  the  Square.  Erect  a 
perpendicular  with  it,  as  above  directed.  Then  sight  along 
the  perpendicular,  and  if  the  original  line  appears  perpen- 
dicular, the  instrument  is  correct ;  if  it  does  not,  the  devia- 
tion will  equal  twice  the  error  of  the  instrument.  Set  a 
pole  in  the  true  perpendicular,  which  will  be  found  as  in 
Art.  224,  and  alter  the  position  of  the  glass  until  the  re- 
flected image  appears  in  the  proper  position.  One  end  of 
the  glass  should  be  movable  by  screws  or  by  little  wedges, 
so  as  to  allow  of  its  position  being  rectified. 


130 


CHAIN  SURVEYING. 


[CHAP.  IV. 


C.— PAEALLELS. 

Problem  1. — Through  a  given  point  to  run  a  parallel  to  a 
given  accessible  line. 


Fig.  91. 


227.  This  may  be  done  by  Arts. 
97,  98,  or  99,  or  thus :—  —  -4 2 

Let  AB  (Fig.  91)  be  the  line,  and  \,$ 

C  the  point.     From  C  to  any  point  /    \x 

D  in  AB,  run   out  the   line  CD.     -g- ^ 

From  E,  any  point  in  CD,  run  a 

line  cutting  AB  in  F.     Then  make  EG  a  fourth  proportional 

EF.EC 

-,  and  GC  will  be  paral- 


to  DE,  EF,  and  EC,  or  EG  = 
lei  to  AB. 


ED 


Problem  2. — To  draw  a  parallel  to  an  inaccessible  line,  two 
points  of  lohich  are  visible. 

228.  Let  AB  (Fig.  92)  be  the 
straight  line,  and  C  the  given 
point.  Run  the  line  CD  per- 
pendicular to  AB,  by  Art.  220 ; 
and  from  C  set  out  CE  perpen- 
dicular to  CD.  It  will  be  the  E~~  c 
parallel  required. 

Problem  3. — To  draw  a  parallel  to  a  given  line  through  an 
inaccessible  point. 


229.  Let  AB  (Fig. 
93)  be  the  given  line, 
and  C  the  given  point. 
From  A,  towards  C, 
run  AC ;  and  in  CA, 
or  CA  produced,  take 
any  point  D.  Run  DE 
parallel  to  AB.  Set 
off  BC  towards  C,  in- 


SEC.  III.]     OBSTACLES  IN  RUNNING  AND  MEASURING  LINES.        131 

tersecting  DE  in  E.     Measure  AB  and  DE.     Run  through 
any  point   in  AB  the  line  BFG-,  intersecting  DE  in  F. 

DE   BF 

Make  FG  =  A  p  '    —  ,  and  CG  will  be  parallel  to  AB. 
AJB  — 


For,  since  FG  =  ——^——>  we  have  AB  -  DE  :  DE  ::  BF  :  FG. 
AB  — 


Whence  AB  :  DE  :  :  BG  :  FG  ; 
but  AB  :  DE  :  :  BC  :  EC  ; 

BG  :  FG  :  :  BC  :  EC,  and  CG  is  parallel  to  EF,  or 
to  AB. 


SECTION  III. 

OBSTACLES  IN  RUNNING  AND  MEASURING  LINES.* 

A.— OBSTACLES  IN  RUNNING  LINES. 

230.  IN  ranging  out  lines  by  the  method  described  in 
Art.  204,  obstacles  are  frequently  met  with  which  prevent 
the  operation  being  directly  carried  on.  In  such  cases 
some  contrivance  is  necessary  in  order  that  the  line  may  be 
prolonged  beyond  such  obstacle.  Various  methods  have 
been  devised  for  this  purpose.  The  following  are  among 
the  most  simple : — 

-231.  First  Method. — By  per-  Fig.  94. 

pendiculars.    Let  AB  (Fig.  94) 
be  the  line,  and  M  the  obsta- 
cle.    At  two  points  C  and  B 
in  AB,  set  off  two  equal  per- 
pendiculars CD  and  BE  long  enough  to  pass  the  obstacle. 
Through  D  and  E  run  the  line  DG ;  and  at  two  points  F 
and   G  beyond   the   obstacle,  set  off  perpendiculars  FH 

*  In  Gillespie's  "Land  Surveying"  may  be  found  a  still  greater  variety  of 
methods  for  these  objects. 


132 


CHAIN  SURVEYING. 


[CHAP.  IV. 


and  GI  equal  to  CD.     Then  HEK  will  be  the  prolongation 
of  AB. 


A  B 


232.  Second  Method.— By 
equilateral  triangles.    Let  AB 
(Fig.  95)  be  the  line,  the 
obstacle  being  at  0.     By 
sweeping  with  the    chain, 
describe  the  equilateral  tri- 
angle BCD.     Prolong  BD 
to  E  sufficiently  far  to  pass 
the  obstacle.    Describe  the 

equilateral  triangle  FEG,  and  prolong  EG  till  EH  =  EB. 
Describe  the  equilateral  triangle  HKI,  and  KH  will  be  the 
prolongation  of  AB. 

233.  Instead  of  making  BEH  an  equilateral  triangle, 
which  would  sometimes  require  the  point  E  to  be  incon- 
veniently remote,  run  BE  (Fig. 

96)  as  before.  Set  out  the  per- 
pendicular EG  =  1.T32  x  BE. 
Describe  the  equilateral  triangle 
GFI.  Bisect  FI  in  H.  Then 
HG  will  be  the  prolongation 
of  BC. 


B.— OBSTACLES  IN  MEASURING  LINES. 

234.  When,  owing  to  any  obstructions,  the  distance  of  a 
line  cannot  be  directly  measured,  resort  should  be  had  to 
trigonometrical  methods.     In  the  absence,  however,  of  the 
proper  instruments,  it  may  be  necessary  to  determine  such 
distances.     The  following  are  a  few  of  the  many  methods 
that  may  be  employed  in  such  cases : — 

1.   To  measure  a  line  when  both  ends  are  accessible. 

235.  Arts.  231,  232,  233,  furnish  means  of  determining 
the  distance  in  this  case.     By  the  method  Art.  231,  BH  = 


SEC.  III.]     OBSTACLES  IN  RUNNING  AND  MEASURING  LINES.        133 

EF ;  and  in  that  of  232,  BH  =  BE.     If  the  method  Art.  233 
is  employed,  BO  =  2  BE. 


2.  When  one  end  is  inaccessible. 

236.  First  Method.— Eun  BE  (Fig.  97) 

in  any  direction,  and  AD  parallel  to  it. 

Through   any  point  D   in  AD,  run  DE 

towards  C.     Measure  AD,  AB,  and  BE : 

AB.BE 

then  BC 


Fig.  97. 


237.  Second  Method.— Set  off  AC  (Fig. 
98)  in  any  direction,  and  CD  parallel  to 
AB.  Eun  DE  towards  B.  Measure  AE, 

AF   PT) 

EC,  and  CD :  then  AB  =        *;    • 

CE 


Fig.  98.      B 


238.  Third  Method.—  Set  off  AD  (Fig. 
99)  perpendicular  to  AB,  and  of  any  dis- 
tance. Eun  DC  perpendicular  to  DB. 

OD2 

Measure  DC  and  CA:   then  CB  =  -  , 

OA 


»T> 

orAB 


Fig.  99. 


3.   When  the  point  is  the  intersection  of  the  line  with  another, 
and  is  inaccessible. 


134 


CHAIN  SURVEYING. 


[CHAP.  IV. 


239.  First  Method.— Let 
AB  and  CD  (Fig.  100)  be 
the  lines,  the  distances  of 
which  to  their  intersection 
are  required.  Set  off  DF 
parallel  to  BA,  and  run 
CFA.  Measure  CD,  CF, 
CA,andFD.  Then  BE  = 
BD.DF  BD.DC 


Fig.  100. 


240.  Second  Method.— Through  H,  (Fig.  101,)  any  point 


in  CD,  run  two  lines 
AF  and  BG.  Make 
FH  in  any  ratio  to  HA, 
and  GH  in  the  same 
ratio  to  HB.  Draw 
FGC,  cutting  CD  in 
C.  Measure  FC  and 
HC.  Then  AE  = 

AH.FC 

HE  = 


Fig.  101. 


AH.HC 
FH~ 

4.  When  both  ends  are  inaccessible. 

241.  Let  AB  (Fig.  102)  be  the  in- 
accessible line.  From  ,  any  con- 
venient point  C,  run  the  lines  CA 
and  CB  towards  A  and  B,  and,  by 
one  of  the  preceding  methods,  find 
CA  and  CB.  In  CA  and  CB,  or 
CA  and  CB  produced,  take  E  and  D 

so  that      CE  :  CA  : :  CD  :  CB. 

Measure  DE. 
Then         CE  :  CA  : :  ED  :  AB. 


Fig.  102. 


SEC.  IV.]  KEEPING  FIELD-NOTES.  135 

SECTION  IV. 
KEEPING  FIELD-NOTES, 

242.  THE  operation  next  in  importance  to  that  of  per- 
forming the  measurements  accurately  is  that  of  recording 
them   neatly,  concisely,  and   luminously.      The  first  is  a 
requisite  that  cannot  be  too  much  insisted  on,  not  only 
in  the  first  notes,  but  in  all  the  calculations  and  records 
connected  with   surveying.     A  rough,  careless  mode  of  re- 
cording observations  of  any  kind  generally  indicates  an 
equal  carelessness  in  making  them.     Carelessness  in  a  sur- 
veyor, on  whose  accuracy  so  much  depends,  is  intolerable. 
Conciseness  is  also  necessary,  but  it  should  never  be  al- 
lowed to  detract  from  the  luminousness  of  the  notes.     By 
this  last  quality  is  meant  the  recording  of  all  the  observa- 
tions in  such  a  mode  as  to  indicate,  in  the  most  clear  man- 
ner, the  whole  configuration  of  the  plat  surveyed,  and  all 
the  circumstances  connected  with  it  which  it  is  intended  to 
preserve.     The  notes  should  be,  in  fact,  a  full  record  of  all 
the  work,  so  as  to  indicate  fully  not  only  what  was  done, 
but  what  was  left  undone. 

243.  First  Method. — By  a  sketch.     The  simplest  mode  of 
recording  the  notes  is  to  draw  a  sketch  of  the  tract  to  be 
surveyed,  on  which  other  lines  can  be  inserted  as  they  are 
measured.     On  this  sketch  may  be  set  down  the  distances 
to  the  various  points  determined. 

When  the  tract  is  large,  however,  or  contains  many  base- 
lines, this  sketch  becomes  so  complicated  as  scarcely  to  be 
capable  of  being  deciphered  after  the  mind  has  been  with- 
drawn from  that  particular  work  and  the  configuration  of 
the  plat  has  been  in  some  measure  forgotten. 

244.  Field-Book.     Perhaps  the  best  kind  of  a  field- 
book  is  one  that  is  long  and  comparatively  narrow,  faint- 
lined  at  moderate  distances.     The  right-hand  page  should 


136  CHAIN  SURVEYING.  [CHAP.  IV. 

be  ruled  from  top  to  bottom  with,  two  lines,  about  an 
inch  apart,  near  the  middle  of  the  page.  The  left-hand 
page  maybe  ruled  in  the  same  manner;  but  it  is  better 
left  for  remarks,  sketches,  and  subsidiary  calculations. 

In  the  space  between  the  vertical  lines  all  the  distances 
are  to  be  inserted:  offsets,  and  other  measurements  con- 
nected with  the  main  line,  may  be  recorded  in  the  spaces  on 
each  side  of  the  column. 

In  recording  the  measurements  the  book  should  be  held 
in  the  direction  in  which  the  work  is  proceeding.  The 
right-hand  side  of  the  column  will  then  coincide  with  the 
right-hand  side  of  the  line,  and  vice  versa.  The  notes 
should  commence  at  the  bottom,  and  all  offsets  and  other 
lateral  distances  must  be  recorded  on  the  side  of  the 
columns  corresponding  to  the  side  of  .the  line  to  which 
they  belong. 

"When  marks  are  left  for  starting  points  for  other  mea- 
surements, the  distance  to  them  should  be  recorded  in  the 
column,  and  some  sign  should  be  made  to  indicate  the 
purpose  for  which  such  distance  was  recorded.  Stations 
of  this  kind  are  called  False  Stations,  and  may  be  desig- 
nated by  the  letters  F.  S. ;  by  a  triangle,  A  ;  or  circle,  o ; 
or  by  surrounding  the  number  by  a  circle,  thus,  f567. ) 
"Whatever  plan  is  adopted  should  be  scrupulously  adhered 
to, — changes  in  the  notation  being  always  liable  to  lead  to 
confusion. 

A  regular  station  may  be  designated  either  by  letters,  A, 
B,  or  by  numbers,  1,  2,  3,  prefixed  by  the  letter  S  or  by  Sta. 
In  the  field-notes  in  the  following  pages  examples  of  most 
of  these  methods  will  be  found. 

Lines  are  referred  to,  either  by  having  them  numbered 
on  the  notes  as  Line  1,  Line  2,  or  by  the  letters  or  figures 
which  designate  the  stations  at  their  ends.  Thus,  a  line 
from  Sta.  1  to  Sta.  3  would  be  referred  to  as  the  line  1,  3 ; 
one  from  Sta.  B  to  Sta.  D,  as  the  line  BD.  This  is  perhaps 
the  best  mode.  Some  surveyors,  however,  refer  to  them  by 
their  lengths.  Thus,  a  line  563  links  long  would  be  called 
the  line  563. 

False  stations  on  a  line  are  named  by  the  line  and  distance. 


SEC.  IV.] 


KEEPING  FIELD-NOTES. 


137 


Thus,  a  station  on  a  line  AB  at  597  links  would  be  called 
F.  S.  597  AB,  orC59T)AB,  or  A,  or  O  597  AB.  It  hardly 
needs  remark,  yet  it  is  of  importance,  that  unity  of  system 
should  be  adopted.  "Whatever  method  of  designating  a 
line  or  station  has  been  employed  in  recording  it,  should  be 
used  in  referring  to  it. 

The  spaces  on  the  right  and  left  of  the  column  will  serve, 
in  addition  to  the  purposes  already  mentioned,  to  contain 
sketches  of  adjoining  lines  and  short  remarks  to  elucidate 
the  work. 

A  fence,  road,  brook,  &c. 
crossing  the  line  measured, 
should  not  be  sketched  as 
crossing  it  in  a  continuous 
line,  as  at  365,  marginal 
plan,  but  should  consist  of 
two  lines  starting  at  opposite  points,  as  at  742,  so  that  if  we 
were  to  suppose  the  lines  forming  the  vertical  column  to 
collapse,  those  representing  the  fence  would  be  continuous. 

When  the  chainmen,  after  closing  the  work  on  one  line, 
begin  the  next  at  the  closing  station,  a  single  horizontal 
line  should  be  drawn;  but  if  they  pass  to  some  other  part 
of  the  tract,  two  lines  should  indicate  the  end  of  the  line. 

To  indicate  the  direction  in  which  a  line  turns,  the  marks 
"1  or  f  may  be  used,  the  former  indicating  that  the  new 
line  bears  to  the  left,  and  the  latter  to  the  right.  Instead 
of  these,  the  words  right  and  left  may  be  used,  or  the  simple 
initials  E.  and  L.  "Whichever  of  the  means  is  used,  the 
sign  should  be  on  the  left  hand  of  the  column  if  the  turn  is 
to  the  left,  and  vice  versd. 


The  following  notes  will  illustrate  all  these  directions! 
They  belong  to  the  tract  Fig.  103. 


138 


CHAIN   SURVEYING. 


[CHAP.  IV. 


Sta.  D 

•^ 

2440 

^>~v 

2020 

^v 

li-^ 

(1395) 

^ 

1 

Sta.  A 

Sta.  A 

1135 

» 

T 

Sta.  C 

Sta.  C 

1760 

__—  -  —  - 

Q50) 

Sta.  B 

Sta.B 

Sl^ 

24^ 

1445 

•\xx'"* 

^ 

1170 

'''^\ 

,*''' 

Sta.  A 

N.45°E. 

Sta.  C 

**^ 

2425 

}_S 

"*** 

1550 

•^^s^ 

t^^. 

1390 

/^J^,  Brook. 

^ 

395 

^ 

Sta.  D 

(1395) 

in  AD 

~~~~'-*^.^ 

1440 

770 

^-'' 

425 

**>fS^*±. 

**"* 

("95JP) 

In  EC  southerly. 

Sta.B 
1760 

^^,  Brook. 

*- 

515 

^r^ 

1 

Sta.  D 

Beginning  at  A,  the  first  line  measured  is  the  diagonal 
AB ;  the  course  N.  45°  E.  is  set  down  at  the  right.  The 
first  point  requiring  notice  is  the  intersection  of  the  dia- 
gonals at  1170  links  from  A.  The  diagonal  is  represented 
by  the  dotted  line  crossing  the  columns,  a  continuous  line 
being  employed  to  designate  a  fence  or  side,  and  a  dotted 
line  a  sight-line.  At  1445  the  fence  EF  is  crossed.  The 
whole  length  to  B  is  2492  links. 


SEC.  IV.]  KEEPING  FIELD-NOTES.  139 

Turning  to  the  left  along  BC,  at  950  we  come  to  the  fence 
bearing  to  the  left:  950  is  surrounded  by  a  line,  thus,  (^  950 >") 
because  it  is  to  be  used  as  a  starting-point  for  another  mea- 
surement. Having  arrived  at  C,  1760  links  from  B,  again 
turn  to  the  left  towards  A:  the  distance  CA  is  1135  links. 
AD  is  next  measured.  At  1395  the  fence  EF  is  found :  the 
point  is  marked  (fT395~)  :  at  2020  the  brook  is  crossed,  and 
at  2440  links  we  find  the  corner  D.  Turning  to  the  left 
along  DB,  at  515  the  brook  is  again  crossed.  This  line  is 
1760  links  long. 

Passing  now  to  E,  f  950  J  in  BC,  along  the  cross  fence, 
the  diagonal  AB  is  passed  at  425;  at  770  CD  is  passed; 
1440  links  brings  us  to  1395  in  AD.  Passing  to  D :  along 
DC,  at  395  the  brook  is  crossed ;  at  1390  the  fence  is  found; 
at  1550  we  cross  the  diagonal  AB:  2425  brings  us  to  C, 
which  finishes  the  work. 

245.  Test-lines.    In  the  above  survey  more  lines  have 
been  measured  than  are  absolutely  necessary.     It  is  always 
better  to  measure  too  many  than  too  few.     If  the  redundant 
lines  are  not  needed  in  the  calculation,  they  serve  as  tests  by 
which  to  prove  the  work.    Tor  the  mere  purpose  of  calcula- 
tion, one  of  the  diagonals  and  the  line  EF  might  have  been 
omitted :  the  other  lines  afford  sufficient  data  for  making  a 
plat  and  calculating  the  area.    An  error  in  one  of  the  others 
will  not  prevent  the  notes  from  being  platted,  and  hence 
they  do  not  in  any  way  afford  a  criterion  by  which  we  can 
judge  of  the  accuracy  of  the  measurements;  but  when  to 
these  are  added  the  length  of  the  other  diagonal  we  have  a 
series  of  values,  all  of  which  must  be  correct  or  the  map 
cannot  be  made. 

246.  General  Directions.    When  about  to  survey  a 
tract  by  this  method,  the  surveyor  should  first  examine  the 
tract  carefully  and   erect  poles  at  the  prominent  points, 
corners,  and  false  stations,  along  the  boundary  lines.     He 
should  stake  out  all  diagonals  and  subsidiary  lines  which 
he  may  wish  to  measure,  setting  a  stake  at  the  points  in 


140  CHAIN  SURVEYING.  [CHAP.  IV. 

which  such  lines  intersect  each  other  or  cross  the  former 
lines, — in  fact,  at  every  point  the  position  of  which  it  may 
be  desirable  to  fix  on  the  plat. 

Having  made  these  preparations,  he  may,  if  the  tract  is 
at  all  complicated,  make  an  eye-sketch.  This  will  serve  to 
guide  him  in  regard  to  the  best  course  to  take  in  his 
measurements. 

Commencing  then  at  some  convenient  point  of  the  tract, 
he  should  measure  carefully  the  diagonals  and  sides  in  suc- 
cession, passing  from  one  line  to  such  other  as  will  make 
the  least  unnecessary  walking,  and  setting  down  in  his  note- 
book the  distance  to  every  stake,  fence,  brook,  or  other  im- 
portant object  met  with. 

When  the  tract  is  large,  the  work  may  last  through 
several  days.  In  such  cases,  each  day's  work  should,  if 
possible,  be  made  complete  in  itself, — that  it  may  be  platted 
in  the  evening.  This  will  prevent  the  accumulation  of 
errors  which  might  occur  from  a  mismeasurement  of  one 
of  the  earlier  lines. 

247.  Platting  the  Survey.  To  plat  a  survey  from  the 
notes,  select  three  sides  of  a  triangle  and  construct  it. 
Then,  on  the  sides  of  this  construct  other  triangles,  until 
the  whole  of  the  lines  are  laid  down.  Measure  test-lines  to 
see  whether  the  work  is  correct. 

In  all  cases  commence  with  large  triangles,  and  fill  up 
the  details  as  the  work  proceeds. 


SEC.  V.]      SURVEYING  FIELDS  OF  PARTICULAR  FORMS.  141 


SECTION  V. 

ON  THE  METHOD  OF  SURVEYING  FIELDS  OF  PAR- 
TICULAR FORMS, 

248.  Rectangles.    MEASURE  two  adjacent  sides:  their 
product  will  give  the  area. 

EXAMPLES. 

Ex.  1.  Let  the  adjacent  sides  of  a  rectangular  field  be 
756  and  1082  links  respectively,  to  plat  the  field  and  calcu- 
late the  content. 

Calculation. 

Content  =  1082  x  756  =  817992  square  links  =  8  A.,  OR., 
28.7  P. 

Ex.  2.  The  adjacent  sides  of  a  rectangular  tract  are  578 
and  924  links :  required  the  area. 

Ans.  5  A.,  1R,  14.51  P. 

Ex.  3.  Required  the  area  of  a  tract  the  sides  of  which 
are  9.75  and  11.47  chains  respectively. 

Ans.  11  A.,  0  R.,  29  P. 

249.  Parallelograms.     Measure  one  side  and  the  per- 
pendicular distance  to  the  opposite  side.     Their  product 
will  be  the  area. 

If  a  plat  is  required,  a  diagonal  or  the  distance  from  one 
angle  to  the  foot  of  the  perpendicular  let  fall  from  the  adja- 
cent angle  may  be  measured. 

EXAMPLES. 

Ex.  1.  Given  one  side  of  a  parallelogram  10.37  chains, 
and  the  perpendicular  distance  from  the  opposite  side  7.63 
chains,  the  distance  from  one  end  of  the  first  side  to  the 
perpendicular  thereon  from  the  adjacent  angle  being  2.75 
chains.  Required  the  area  and  plat. 

Ans.  7  A.,  3  R.,  25.97  P. 


A42  CHAIN  SURVEYING.  [CHAP.  IV. 

Ex.  2.  Desiring  to  find  the  area  of  a  field  in  the  form  of 
a  parallelogram,  I  measured  one  side  763  links,  and  the 
perpendicular  from  the  other  end  of  the  adjacent  side  647 
links,  said  perpendicular  intersecting  the  first  side  137  links 
from  the  beginning.  Required  the  content  and  plat. 

Ans.  4  A.,  3  E.,  29.86  P. 

250.  Triangles.    First  Method. — Measure  one  side,  and 
the  perpendicular  thereon  from  the  opposite  angle ;  noting, 
if  the  plat  is  required,  the  distance  of  the  foot  of  the  per- 
pendicular from  one  end  of  the  base. 

Multiply  the  base  by  the  perpendicular,  and  half  the  pro- 
duct will  be  the  area. 

EXAMPLES. 

Ex.  1.  Required  the  area  and  plat  of  a  triangular  tract, 
the  base  being  7.85  chains  and  the  perpendicular  5. 47  chains, 
the  foot  of  the  perpendicular  being  3.25  chains  from  one 
end  of  the  base. 

Calculation. 

7.85x5.47       42.9395         . 
Area  = — =  — =  21.46975  chains  =  2  A., 

0  R.,  23.5  P. 

Ex.  2.  Required  the  area  and  plat  of  a  triangle,  the  base 
being  10.47  chains,  and  the  perpendicular  to  a  point  4.57 
chains  from  the  end,  being  7.93  chains. 

Ex.  3.  Required  the  area  of  a  triangle,  the  base  being 
1575  links,  and  the  perpendicular  894  links. 

251.  Second  Method. — Measure  the  three  sides,  and  calcu- 
late by  the  following  rule: — 

From  half  the  sum  of  the  sides  take  each  side  severally;  mul- 
tiply the  half-sum  and  the  three  remainders  continually  together, 
and  the  square  root  of  the  product  will  be  the  area. 


SEC.  V.]        SURVEYING  FIELDS  OF  PARTICULAR,  FORMS. 


143 


DEMONSTRATION. — Let  ABC  (Fig.  104)  be  Fig.  104. 

a  triangle.  Bisect  the  angles  C  and  A  by 
the  lines  CDH  and  AD,  cutting  each  other 
in  D.  Then  D  is  the  centre  of  the  inscribed 
circle.  Join  DB,  and  draw  DE,  DP,  and 
DG  perpendicular  to  the  three  sides.  Then 
will  DE  =  DF  =  DG,  and  (47.1)  FB  =  BG, 
CE  =  CF,  and  AE  =  AG. 

Bisect  the  exterior  angle  KAB  by  the 
line  AH,  cutting  CDH  in  H.  Draw  HK, 
HL,  and  HM  perpendicular  to  CA,  AB, 
and  CB.  Join  HB.  Then  (26.1)  KH  = 
HM,  CK  =  CM,  HL  =  HK,  and  AL  =  AK ; 
also  (47.1)  BL  =  BM.  Because  AK  =  AL 

and  BM  =  BL,  CK  -f  CM  will  be  equal  to  the  sum  of  the  sides  AB,  AC,  and 
BC ;  therefore  CK  or  CM  =  $  (AB  -f  AC  -f  BC)  =  £  S,  if  S  stand  for  the 
sum  of  the  three  sides.  But  CE  -f  AE  +  BG  =  £  S  ;  therefore  CK  =  CM  = 
CA  +  BG,  and  AK  =  AL  =  BG;  whence  AG  ==  AE  =  BL  =  BM,  and  EK  = 
AB.  Now,  since  CK  =  CM  =  J  S,  we  have  AK  =  £  S  —  AC,  EC  =  £  S  —  AB, 
and  AE  =  BM  =  J  S  —  BC. 

Because  the  triangles  CDE  and  CKH,  as  also  ADE  and  HKA,  are  similar, 


we  have  (4.6) 
and 

(23.6) 
Whence, 
and 


CE  :  ED  :  :  CK  :  KH, 
AE  :  ED  : :  HK  :  KA, 
AE  .  EC  :  EDa  :  :  CK  :  KA  : :  CKa  :  CK .  KA. 


.  EC  :  ED  :  :  CK  :  ^/CK  .  KA, 


CK  .  ED  ==  v'CK .  KA  .  AE  .  EC. 


Now,  ABC  =  ACD-f-  BCD  -f  ABD 
S  .  ED  =  CK  .  ED. 


AC  .  ED-f  £  BC  .  ED  +  %  AB  .  ED 


Wherefore,        ABC  =  ^/CK  .  KA.  AE  .  EG. 

COR.  —  From  the  above  demonstration,  it  is  apparent  that  the  area  of  a  tri- 
angle is  equal  to  the  rectangle  of  the  half-sum  of  the  sides  and  the  radius  of 
the  inscribed  circle. 

For  another  demonstration  of  this  rule,  see  Appendix. 


EXAMPLES. 

Ex.  1.  Eequired  the  area  of  a  triangle,  the  three  sides 
being  672,  875,  and  763  links  respectively. 

NOTE. — In  cases  of  this  kind  the  operation  will  be  much  facilitated  by  using 
logarithms. 


144  CHAIN  SURVEYING.  [CHAP.  IV. 

672  +  8T5  +  763       2310 

=  — - —  =  1155  =  half-sum  of  sides. 

2i  L 

J  sum     =  1155  log.  3.062582 

J  sum  —  672  =  483  log.  2.683947 

\  sum  —  875  =  280  log.  2.447158 

\  sum  —  763  =  392  log.  2.593286 

2)10.786973 

Area,      247449  square  links,  5.393486 
=  2  A.,  1  K.,  35.9  P. 

Ex.  2.  Eequired  the  area  of  a  triangular  tract,  the  sides 
of  which  are  17.25  chains,  16.43  chains,  and  14.65  chains 
respectively.  Ans.  11  A.,  0  R.,  14.4  P. 

Ex.  3.  Given  the  three  sides,  19.58  chains,  16.92  chains, 
and  12.76  chains,  of  a  triangular  field :  required  the  area. 

Ans.  10  A.,  2  R.,  27  P. 

252.  Trapezoids.  Measure  the  parallel  sides  and  the  per- 
pendicular distance  between  them. 

If  a  plat  is  desired,  a  diagonal,  or  the  Fis- 105- 

distance  AE,  (Fig.  105,)  may  be    mea- 
sured. 


Multiply  the  sum  of  the  parallel  sides  by     A 
half  the  perpendicular :  the  product  is  the  area. 

DEMONSTRATION.  —  ABCD  ==  ABD  +  BCD  =  $  AB  .  DE  +  $  DC  .  DE  == 
(AB-f  DC).  £DE. 

EXAMPLES. 

Ex.  1.  Given  AB  =  7.75  chains,  DC  =  5.47  chains,  and 
DE  =  4.43  chains,  to  calculate  the  content  and  plat  the 
map,  AC  being  7.00  chains. 

Ans.  Area,  2  A.,  3  R.,  28.5  P. 

Ex.  2.  Given  the  parallel  sides  of  a  trapezoid,  16.25  chains 
and  14.23  chains,  respectively:  the  perpendicular  from  the 
end  of  the  shorter  side  being  12.76  chains,  and  the  distance 


SEC.  V.]        SURVEYING  FIELDS  OF  PARTICULAR  FORMS.  145 

from  the  foot  of  said  perpendicular  to  the  adjacent  end  of 
the  longer  side  1.37  chains.     Required  the  area  and  plat. 

Ans.  19  A.,  1  E.,  31.4  P. 

'253.  Trapeziums.  First  Method. — Measure  a  diagonal, 
and  the  perpendiculars  thereon,  from  the  opposite  angle. 

The  area  of  a  trapezium  is  equal  to  the  rectangle  of  the 
diagonal  and  half  the  sum  of  the  perpendiculars  from  the 
opposite  angles. 

This  is  evident  from  the  triangles  of  which  the  trapezium 
is  composed. 

EXAMPLES. 

Ex.  1.  To  plat  and  calculate  the  area  of  a  trapezium,  the 
diagonal  being  15.63  chains,  and  the  perpendiculars  thereto 
from  the  opposite  angles  being  8.97  and  6.43  chains,  and 
meeting  the  diagonal  at  the  distances  of  4.65  and  13.23 
chains.  Ans.  Area,  12  A.,  0  R.,  5.6  P. 

Ex.  2.    Given  (Fig.  106)  AC  =  19.68  Fig.  i06. 

chains,  AE  =  7.84  chains,  AP  =  16.23 
chains,  ED  =  10.42  chains,  and  FB  = 
8.73  chains,  to  plat  the  figure  and  find 
the  area. 

Ans.  18  A.,  3  R.,  14.98  P. 

Ex.  3.  Required  the  area  of  a  trape- 
zium, the  diagonal  being  17.63  chains,  and  the  perpen- 
diculars 6.47  and  12.51  chains  respectively. 

Ans.  16  A.,  2  R.,  36.94  P. 

254.  Second  Method. — Measure  one  side,  and  the  perpen 
diculars  thereon  from  the  extremities  of  the  opposite  side, 
with  the  distances  of  the  feet  of  these  perpendiculars  from 
one  end  of  the  base. 


10 


146 


CHAIN  SURVEYING. 


[CHAP.  IV. 


Fig,  107. 
C 


EXAMPLES. 

Ex.  1.  Let  ABCD  (Fig.  107) 
be  a  trapezium,  of  which  the  fol- 
lowing dimensions  are  given, — 
viz. :  AE  =  3.27  chains,  AF  = 
10.17  chains,  AB  =  17.62  chains, 
ED  =  7.29  chains,  and  FC  = 
13.19  chains.  Required  to  plat 
it,  and  calculate  the  area. 

Lay  off  the  distances  AE,  AF,  and  AB ;  then  erect  the 
perpendiculars  ED  and  FC,  and  draw  AD,  DC,  and  CB. 

The  trapezium  is  divided  into  two  triangles  and  the 
trapezoid,  the  areas  of  which  may  be  found  by  the  pre- 
ceding rules. 

Thus,  2AED=  AE.ED  =    23.8383 

2  EFCD  =  EF.(ED  +FC)=  141.3120 
2  CFB  =  CF.  FB  =    98.2655 

whence    ABCD  =  J  of  263.4158  =  131.7079 

chains  =  13  A.,  0  E.,  27.3  P. 

If  either  of  the  angles  A  or  B  were  obtuse,  the  perpen- 
dicular would  fall  outside  the  base,  and  the  area  of  the 
corresponding  triangle  should  be  subtracted. 

Ex.  2.  Plat  and  calculate  the  area  of  a  trapezium  from 
the  following  field-notes : — 


OB 

1143 

perp.  936 

917 

perp.  825 

415 

0  A 

Ans.  7  A.,  0  E.,  30.3  P. 

Ex.  3.     Calculate  the    area  from  the  following    field- 
notes  : — 


Ans.  6  A.,  2  E.,  2  P. 


perp.  892 

1365 

967 

Stat.  B. 

perp.  568 

376 

0  A 

SEC.  V.]        SURVEYING  FIELDS  OF  PARTICULAR  FORMS. 


147 


Fields  of  more  than  four  sides,  bounded  by 
straight  lines. 

255.  First  Method. — Divide  the  tract  into  triangles  and 
trapeziums,  and  calculate  the  areas  by  some  of  the  pre- 
ceding rules.  In  applying  this  method,  as  many  of  the 
measurements  as  practicable  should  be  made  on  the 
ground ;  the  field  then  being  platted  with  care,  the  other 
distances  may  be  measured  on  the  map.  When  it  is 
intended  to  depend  on  the  map  for  the  distances,  every 
part  of  the  plat  should  be  laid  down  with  scrupulous  ac- 
curacy, on  a  scale  of  not  less  than  three  chains  to  the 
inch. 

Ex.  1.  To  draw  the  map  and  calculate  from  the  follow^ 
ing  field-notes  the  area  of  the  pentagonal  field  ABODE : — 


oD 

1 

690 

1 

570 

510  C 

1  E.  350 

280 

OA 

N.15°E. 

0  C 

oC 

1 

770 

4 
a 

915 

1 

510 

250  B     z 

585 

Brook. 

! 

360 

Brook.     H 

365 

AD 

©A 

E.  of  AD 

o  E 

The  construction  is  plain. 

Calculation. 

Twice  trapezium  ACDE  =  AD 
x  (Ea  +  60)  =  6.90  x  8.60  = 
59.34;  twice  triangle  ABC  = 
AC  x  Be  =  7.70  x  2.50  =  19.25; 


Fig.  108. 


=  39.295  ch.  =  3  A.,  3  R,  28.72  P. 


Ex.  2.  Map  the  plat,  and  calculate 
the  content  from  the  following  field- 
notes  : — 


Fig.  109. 


148 


CHAIN  SURVEYING. 


[CHAP.  IV. 


0D 

520 

288 

80  E 

G120 

206 

0  F 

©G 

440 

D230 

150 

00 

Lof  CA 

©c 

550 

B180 

410 

135 

130  G 

©A 

East. 

Construction. 

Commencing  at  A,  (Fig.  109,)  draw  the  line  AC  east 
5.50  chains,  marking  the  points  a  and  b  at  1.35  and  4.10 
chains  respectively :  at  a  and  b  erect  the  perpendiculars  aG 
1.30  and  £B  l.SO  chains.  From  C  to  G  draw  CG,  which 
should  be  4.40  chains  long.  At  c,  1.50  chains  from  C, 
draw  cD  perpendicular  to  CG  and  equal  to  2.30  chains. 
With  the  centre  G  and  radius  1.20  chains,  describe  a  circle, 
and  from  D  draw  the  line  DF  5.20  chains  long,  touching 
the  circle  at  e,  which  should  be  2.06  chains  from  F.  At  d, 
2.88  chains  from  F,  draw  the  perpendicular  dE  —  .80  chains: 
then  will  A  B  C  D  E  F  G  be  the  corners  of  the  tract. 

Calculation. 

2  ABCG  =  AC  (Ga  +  B6)  =  5.50  x  3.10  =  1T.05; 
2   GCD    =  GC  .  cD  =  4.40  x  2.30  =  10.12; 

2  GDEF  =  FD  (Ge  +  <2E)  =  5.20  x  2.00  =  10.40. 


Therefore  area 
3  E.,  20.56  P. 


37.5T    ,    . 
— - —  chains 


18.785  chains  =  1  A., 


Ex.  3.  Required  the  plans  and  areas  of  the  adjoining 
fields,  of  which  the  following  are  the  field-notes,  the  two 
fields  to  be  platted  on  one  map. 


SEC.V.]        SURVEYING  FIELDS  OF   PARTICULAR  FORMS. 


149 


0(4) 
970 

(3)  772 

830 

395 

284  (5) 

0(6) 

KE. 

(2)  395 

320 

100 

0(1) 

715  (6) 
K  10°  E. 

Area  10  A.,  2  E.,  18.576  P. 


o  7 
1150 
675 

0(8) 

432  (11) 

(8)  565 

0(9) 
1285 
1000 
960 
0(7) 

155  (10) 
L.  of  (7,5) 

(4)  562 

ISi 

390 

282 
0(5) 

313  (10) 
R.  of  (4) 

Area  12  A.,  3  R.,  18.1  P. 

Ex.  4.  Required  the  plan  and  areas  of  the  adjoining  fields 
from  the  following  field-notes,  tracing  thereon  the  course 
of  the  brooks. 


0(7) 

1051 

Brook  +  (6.7)— 

680 

-v^^ 

648 

540  (1) 

~N~~^-^ 

510 

•N^^^  Brook. 

365 

—Brook  +  (1.5) 

(6)  380 

130 

0(5) 

r 

0(5) 

1255 

853 

T65  (1) 

(4)  500 

440 

• 

0(3) 

r 

0(3) 
1150 

& 

Brook  +  (2.3)— 

850 

VN'N-\-V- 

490 

-v_.      Brook. 

(2)  482 

200 

^^s 

0(1) 

(11)  620 

0(10) 
1080 
730 

0(8) 

465  (9) 

KE. 

(6)  665 

0(11) 
1395 
1095 
748 
325 
270 
55 
0(7) 

—Brook  +  (8.11) 

/635  (8) 

\  —  Junction 
^-N-^  of  Brooks. 
—Brook  +  (7.8; 

R.  of  (7.5) 

Area  14  A.,  3R.,  28.24  P. 


Area  15  A.,  2  R.,  7  P. 


NOTE. — In  the  above  field-notes  the  marginal  references,  such  as  "  Brook  -j 
6.7,"  means  to  the  point  in  which  the  brook  crosses  the  line  (6.7.) 


150 


CHAIN   SURVEYING. 


[CHAP.  IV. 


256.  Second  Method. — Instead  of  running  diagonals,  it  may 
sometimes  be  more  convenient  to  run  one  or  more  lines 
through  the  tract  and  take  the  perpendiculars  to  the  several 
angles,  as  in  the  following  example. 

Let  the  field  be  of  the  form 
ABCDEF,  (Fig.  110.)  Run  the  line 
AC,  and  take  the  perpendiculars  /F, 
eE,  £B,  and  dD.  The  field  will  thus 
be  divided  into  triangles  and  trape- 
zoids,  the  area  of  which  may  be 
calculated  by  the  preceding  rules. 

Thus,  let  the  field-notes  of  the  preceding  tract  be  as 
follows : — 


0C 

1185 

D420 

840 

760 

200  B 

E280 

590 

F220 

250 

0A 

East. 

Dist. 

Perp. 

Int. 
Dist. 

Sum  of 
Perp. 

Double 

Areas. 

2  AFf 
2  /FE« 
2eEDd 
2DdC 

Left-hand  areas. 
Eight   "       " 

0 

250 
590 
840 
1185 

0 
220 

280 
420 
0 

250 

340 
250 
345 

220 
500 
700 
420 

55000 
170000 
175000 
144900 

1185  x  200 

544900 
237000 

2)  781900 

39.0950  ch.  =  3  A.,  3  E.,  25.5  P. 

The  calculation  being  performed  thus: — In  the  first 
column  are  placed  the  distances  to  the  feet  of  the  left-hand 
perpendiculars.  In  the  second  the  perpendiculars  them- 
selves. The  numbers  in  the  third  column  are  found  by 
subtracting  each  number  in  column  1  from  the  succeeding 
number  in  the  same  column.  The  numbers  in  column  3 


SEC.V.]       SURVEYING  FIELDS   OF  PARTICULAR  FORMS. 


151 


therefore  represent  the  distances  A/,  fe,  ed,  and  dC.  The 
numbers  in  the  fourth  column  are  found  by  adding  each 
number  in  column  2  to  the  succeeding  number  in  the 
same  column ;  they  therefore  are  the  sums  of  the  adjacent 
perpendiculars.  Those  in  the  fifth  column  are  found  by 
multiplying  the  corresponding  numbers  in  columns  3  and 
4.  They  therefore  are  the  double  areas  of  the  several 
trapezoids  and  triangles. 

Ex.  2.  Required  to  calculate  the  content  and  make  plats 
from  the  following  field-notes : — 


1 

oG 

312T 

2590 

476  F 

H3T5 

2145 

20TO 

642  E 

1400 

1920 

1485 

523  D 

840 

5160 

K600 

790 

200 

465  B 

0  A 

E. 

0  F 

4025 

3617 

792  G 

3254 

826  H 

E594 

2846 

D435 

2137 

1548 

3191 

C729' 

1026 

429 

623  K 

B237 

175 

0A 

K  15°  E. 

Area  25  A.,  1  R,  5  P. 


Area  38  A.,  3  R,  17.5  P. 


257.  Offsets.  In  what  precedes,  the  sides  have  been  sup- 
posed to  be  right  lines.  This,  is  ordinarily  the  case  except 
when  the  tract  bounds  on  a  stream.  It  then,  if  the  stream 
is  not  navigable,  generally  takes  in  half  the  bed.  Lands 
bounding  on  tide-water  go  to  low-water  mark.  In  all  such 
cases  the  area  and  configuration  of  the  boundary  are  most 
readily  determined  by  offsets. 

A  base  is  run  near  the  crooked  boundary,  and  perpen- 
dicular offsets  are  taken  to  the  line,  whether  it  be  the  middle 
of  the  stream  or  low- water  mark.  If  the  positions  of  these 
offsets  are  judiciously  chosen,  so  that  the  part  of  the  boun- 
dary intercepted  between  any  two  consecutive  ones  is  nearly 
straight,  the  correct  area  may  be  calculated  precisely  as  in 
last  article.  In  the  field-notes  the  distances  are  written  in 
the  column  and  the  offsets  on  the  right  or  left  hand,  accord- 
ing as  they  are  to  the  right  or  left  of  the  line  run. 


152 


CHAIN  SUEVEYING. 


[CHAP.  IV. 


Thus,  supposing  it  were 
required  to  find  the  area 
contained  between  the  line 
AB  and  the  stream,  (Fig. 
Ill,)  the  following  being 
the  field-notes. 


Fig.  111. 


©B 

25 

865 

70 

725 

165 

580 

165 

475 

100 

355 

115 

195 

90 

75 

40 

0 

©A 

K10°E. 

The  calculation  would  be  as  below,  the  same  formula 
being  used  as  in  last  article. 


Dist. 

Offs. 

Int. 
Dist. 

Sum  of 
Offs. 

Double 

Areas. 

0 

40 

75 

90 

75 

130 

9750 

195 

115 

120 

205 

24600 

355 

100 

160 

215 

34400 

475 

165 

120 

265 

31800 

580 

165 

105 

330 

34650 

725 

70 

145 

235 

34075 

865 

25 

140 

95 

13300 

2)  182575 

Area  =  3  E.,  26  P.     9.12875  ch. 

SEC.V.]        SURVEYING  FIELDS  OP  PARTICULAR  FORMS. 


153 


Ex.  1.  Eequired  the  area  and  plan  from  the  following 
notes : — 


\ 

A 

4830 
2040 
F 

r  ^  eo 

E 
1471 

930 

485 
0 
D 

/ 

fs' 

D 

5000 
3585 
G 

/ 

E355 

F 
2175 
1428 
D 

95 
140 
60 

r 

r 

A 

3000 
Q 

r 

C665 
55 

270 
396 

310 
340 
50 

D 
4175 
3335 

1929 
1408 

1015 
610 
0 

A 

on  creek-bank. 

60 

130 

B      85 
55 

D 

1072 
750 

390 

0 
C 

G 
4241 
F 

r 

r 

F 

826 
420 
0 

E 

r 

55 
55 

K56J°E.  1 

c 

1350 
0 

(2160) 

75 
100 
60 

B  on  A.D. 

Fig.  112  is  a  plat  of  this  tract. 


154 


CHAIN  SURVEYING.                            [CHAI 

Calculation. 

To  find  AGF,  Art. 

251. 

AG 

3000 

F& 

4241 

FA 

4830 

2)12071 

i 

sum 

6035.5 

3.780713 

i 

s-AG 

3035.5 

3.482230 

i 

s-FG 

1794.5 

3.253943 

i 

5  -AF 

1205.5 

3.081167 

2)13.598053 

AGF 

=          6295435 

6.799026 

To  find  AFD. 

AF 

4830 

AD 

4175 

FD 

2175 

2)11180 

i 

sum 

5590 

3.747412 

i 

5-AF 

760 

2.880814 

i 

s-AD 

1415 

3.150756 

i 

s  —  FD 

3415 

3.533391 

2)13.312373 

AFD 

=         4530917 

6.656186 

SEC.  V.]        SURVEYING  FIELDS  OF  PARTICULAR  FORMS. 

To  find  BCD. 


155 


BC 

1350 

BD 

2015 

CD 

1072 

i 

sum 

2)4437 

3.346059 

2218.5 

1 

s-BC 

868.5 

2.938770 

i 

s-BD 

203.5 

2.308564 

i 

5  -CD 
BCD  = 

1146.5 
670475 

3.059374 

2  )  11.652767 

5.826383 

To  find  DEF. 

DE 

1471 

EF 

826 

J 

1 

DF 

sum 
s  -DE 

2175 

3.349472 

2.883661 

2)4472 
2236 
765 

I 

s-EF 

1410 

3.149219 

I 

5-DF 

61 

1.785330 

2)11.167682 

DEF 


383567 


5.583841 


156 


CHAIN  SURVEYING. 


[CHAP.  IV. 


Base. 

Dist. 

Offsets. 

Inter. 
Dist. 

Sum  of 
Offsets. 

Double 
Areas. 

AB 

0 
610 
1015 
1408 
1929 
2160 

50 
340 
310 
396 
270 
55 

610 
405 
393 
521 
231 

390 
650 
706 
666 
325 

237900 
263250 

277458 
346986 
75075 

BO 

1350 

110 

148500 

CD 

0 
390 
750 
1072 

55 

85 
130 
60 

390 
360 
322 

140 
215 
190 

54600 
77400 
61180 

DE 

0 
485 
930 
1471 

60 
140 
95 

60 

485 
445 
541 

200 
235 
155 

97000 
104575 

83855 

EF 

0 
420 

826 

60 
100 

75 

420 

406 

160 
175 

67200 
71050 

2 ) 1966029 

Area  of  part  cut  off  by  bases,     983014.5 

AGF  6295435 

AFD  4530917 

BCD  670475 

*     DEF  383567 


128  A.,  2  R.,  21.5  P. 


12863408.5  links. 


SEC.  V.]        SURVEYING  FIELDS  OF  PARTICULAR  FORMS. 


157 


The  field-notes  of  a  meadow,  bounding  on  a  river  and 
divided  into  four  fields,  are  as  follows, — the  measurements 
being  to  low-water  mark.  Required  the  map  and  the 
content  of  the  whole: — 


55 

72 
97 

75 


D 


cq 

CO 


1054 

896 

739 

480 

C 


C 

<M 

CO 


1622 

1081 

B 


E 

CD 

Oi 

CO 

TH 

585 

-x^ 

B 

r 

B 

o 

TH 

63 

1414 

35 

1237 

87 

1016 

45 

824 

50 

652 

551 

^-~^^ 

452 

75 

295 

75 

D 

toB 

D 

cq 

cb 

0 

CN 

Diagonal. 



1310 



.X 

992 

^^ 

/ 

A 

toD 

o 

US) 

TH 

1030 

^^ 

A 

r^ 

A 

TH 

752 

E 

r 

S71°E 


Area,  34  A.,  3  R. 


To  find  the  contents  of  the  several  enclosures,  other  lines 
would  be  required :  these  might  be  measured  on  the  plat, 
if  it  were  drawn  with  neatness  and  accuracy. 


158  CHAIN  SURVEYING.  [CHAP.  IV. 

SECTION  VI. 
TIE-LINES, 

258.  Tie-Lines.  The  external  boundaries  of  a  tract  of 
land  having  more  than  three  sides  are  not  sufficient  either 
for  making  a  plat  or  calculating  the  area.  In  the  methods 
heretofore  laid  down,  diagonals  were  also  used.  In  some 
cases,  however,  owing  to  obstructions,  such  as  ponds,  close 
woods,  or  buildings,  it  is  difficult  to  run  the  diagonals. 
When  this  is  the  case,  a  line  measured  across  one  of  the 
angles  of  a  quadrilateral  will  determine  the  direction  of  two 
sides,  and  thus  fix  the  relative  position  of  all  the  lines  of  the 
tract.  Such  lines  are  called  tie-lines. 

For    example,    suppose    it  pig.  113. 

were  required  to  survey  the 
tract  represented  in  Fig.  113, 
the  interior  of  which  is  filled 
with  such  thick  woods  that 
the  diagonals  cannot  be  mea- 
sured :  the  external  lines  AB, 
BC,  CD,  and  DA  might  be 
measured  as  before.  Then  /  ...-'''' 

on  the  lines  adjacent  to  one  /  /"' 

angle,  as  C,  measure  carefully  iT 

CE  and  CF ;  also  measure  EF.  These  measures  should  be 
made  with  the  greatest  accuracy,  as  a  slight  error  here  will 
very  materially  affect  the  result.  On  the  same  account,  the 
distances  CE  and  CF  should  be  taken  as  large  as  circum- 
stances will  allow. 

If  the  tie-line  cannot  be  run  within  the  tract,  the  points 
may  be  taken  at  E  and  F  in  the  sides  produced. 

To  plat  such  a  tract,  commence  with  the  triangle.  This 
being  formed,  the  direction  of  CB  and  CD  is  known. 

259.  To  calculate  the  Area.  First  find  in  ECF  the 
angle  ECF,  whence  by  trigonometry  BD  is  found,  and  then 
the  area  of  the  triangles. 


SEC.  VI.]  TIE-LINES.  159 

If  CE  =  CF,  EF  will  be  the  chord  of  the  arc  to  the 

EF 
radius  CE,  whence  the  chord  to  radius  1  =  — — .     This 

quotient  being  found  in  the  table  of  chords  the  correspond- 
ing arc  will  give  the  degrees  and  minutes  of  the  angle  EOF : 
or  CE  :  J  EF  : :  rad.  :  sin.  J  ECF. 

260,  Inaccessible  Areas.    By  a  combination  of  tie-lines 
and  offsets,  tracts  that  cannot  be  entered,  such  as  a  pond  or 
a  swamp,  may  be  measured.     For  this  purpose,  surround 
the  tract  by  a  system  of  lines  bound  at  the  angles  by  tie- 
lines,  and  take  offsets  to  the  prominent  points  in  the  bound- 
ary of  the  tract. 

261.  Defects  of  this  Method.    Every  system  of  measure- 
ment or  drafting  should  commence  with  the  longer  lines 
and  end  with  the  shorter.     By  this  means  the  errors  that 
are  unavoidable  are  diminished  as  we  proceed.     If,  for 
example,  a  diagonal  of  thirty  chains  were  measured,  this 
would  fix  the  distance  of  the  ends  to  a  degree  of  certainty 
precisely  equal  to  that  of  the  measurement ;  and  if  from  this 
measurement  the  length  of  an  inferior  line  joining  two 
points  in  the  sides  were  to  be  determined,  the  errors  in 
the  length  of  the  diagonal  would  affect  this  length  to  a 
degree   exactly  proportional  to  its   length,  the  error  in  a 
line  of  five   chains   long  being   one-sixth  of  that  of  the 
diagonal.     Precisely  the  reverse  is  the  case  when  the  shorter 
line  is  measured :   the  error  is  magnified  as  we  proceed. 
On  this  account,  the  method  explained  above  should  never 
be  employed  when  it  can  be  avoided.     By  the  use  of  the 
compass,  transit,  or  theodolite,  this  can  always  be  done. 
The  mode  of  using  them  for  surveying  purposes  forms  the 
subject  of  the  next  chapter. 


CHAPTER  V. 

COMPASS   SURVEYING. 


SECTION  I. 

DEFINITIONS  AND  INSTRUMENTS. 

262.  IN  chain  surveying,  the  position  of  any  point  is 
determined  either  by  directly  measuring  to  it  from  other 
known  points,  or  by  determining  its  distance  from  such 
points  by  the  indirect  methods  explained  in  last  chapter. 
In  the  method  about  to  be  explained,  its  position  is  ascer- 
tained by  angular  measurements  taken  from  known  stations, 
or  by  its  distance  from  a  known  point  and  the  angle  which 
it  makes  with  the  meridian. 

All  those  methods,  which  have  a  direct  reference  to  the 
meridian  as  the  base  of  angular  distance,  are  known  under 
the  head  of  compass  surveying;  whether  the  instrument 
used  to  determine  the  angle  is  a  theodolite,  a  transit,  or  a 
compass. 

263.  The  Meridian.    If  the  heavens  are  examined  during 
a  clear  night,  the  stars  to  the  north  will  be  perceived  to 
revolve  around  a  star  elevated  about  40°.     This  is  called 
the  pole-star,  and  is  very  nearly  in  the  point  in  which  the 
axis  of  the  earth  if  produced  would   meet  the-  heavens. 
This  point  is  called  the  north  pole  of  the  heavens.     The 
north  star  is  not  exactly  at  the  pole,  but  revolves  around  it 
in  a  small  circle.     If  a  transit  or  theodolite  be  levelled,  and 
the   telescope   directed  to   the   centre  of  this   circle   (see 
chap,  ix.)  it  will  point  exactly  north.     Depress  it,  and  run 

160 


SEC.  I]  DEFINITIONS  AND  INSTRUMENTS.  161 

out  a  line  in  the  direction  of  the  line  of  collimation.     This 
will  be  a  meridian  line. 

264.  The  Points  of  the  Compass.  If  through  any 
station  a  line  be  drawn  perpendicular  to  the  meridian  it  will 
run  east  and  west.  If  we  face  the  south,  the  west  will  be 
to  the  right  hand  and  the  east  to  the  left.  These  four  points — 
north,  east,  south,  west — are  called  the  cardinal  points  of  the 
compass,  and  are  used  as  reference  for  all  angular  distances 
from  the  meridian. 

Fig.  114. 


For  nautical  purposes,  each  of  the  quadrants  into  which 
the  horizon  is  divided  is  further  divided  into  eight  parts 
called  points,  and  named  as  in  Fig.  114,  commencing  at  the 
north  and  going  to  the  east. 

North,  N.;  North  by  East,  (N.frE.;)  North  Northeast, 
(N.N.E. ;)  Northeast  by  North,  (N.E.6N. ;)  Northeast,  (N.E. ;) 
Northeast  by  East,  (N.E.&E.;)  East  Northeast,  (E. N.E. ;)  East 
by  North,  (E.6N.;)  East,  (E.)  and  so  on,  E.6S.;  E.S.E.; 
S.E.6E.;  S.E.;  S.E.68. ;  S.S.E.;  S.fcE.;  S. 

For  land  surveying  only  the  cardinal  points  are  men- 
tioned, the  direction  being  determined  by  the  angular  dis- 
tance from  the  meridian. 

265.  Bearing.  The  bearing  of  a  line  is  the  angle  which 
it  makes  with  a  meridian  through  one  end.  It  is  expressed 
either  by  naming  the  points,  as  N.6E.,  S.S.E.  J  E.,  as  is 

11 


162  COMPASS  SURVEYING.  [CHAP.  V. 

done  iii  navigation,  or  by  mentioning  the  number  of  degrees 
in  the  angle  accompanied  by  the  cardinal  points  between 
which  it  runs.  Thus,  if  a  line  runs  between  north  and  west 
and  makes  an  angle  of  37°  25'  with  the  meridian,  its  bearing 
is  ET.  37°  25'  W.  It  deflects  37°  25'  from  the  north  towards 
the  west,  and  is  therefore  sometimes  said  to  run  from  north 
towards  the  west.  This  expression,  though  convenient,  is 
not  strictly  correct. 

266.  The  Reverse  Bearing.    If  the  bearing  of  a  line 
of  moderate   length  is  determined  at  one  end,  and  then 
again  at  the  other  end,  the  latter  is  called  the  reverse  bearing. 
It  will  be  found  to  be  of  the  same  number  of  degrees  as  the 
bearing,  but  with  the  opposite  points.     Thus,  if  the  bearing 
of  a  line  be  K  27J°  E,  its  reverse  bearing  is  S.  27J°  W. 

If  the  line  be  long,  there  will  be  a  continual  variation 
from  the  initial  course.  Thus,  if  a  line  run  N".  45°  E.  through 
its  whole  course,  it  will  be  found  to  deviate  to  the  left  from 
a  straight  line.  A  true  east  and  west  line  in  latitude  40° 
is  a  curve  with  a  radius  of  about  4800  miles. 

267.  The  Magnetic  Needle.    A  magnetic  needle  is  a 
light  bar  of  magnetized  steel  suspended  on  a  pivot,  so  that 
it  may  turn  freely  in  a  horizontal  direction.     Such  a  needle 
will  always  place  itself  in  nearly  the  same  direction,  one 
end  of  it  being  northward  and  the  other  southward.     The 
needle  should  move  very  freely  on  its  pivot,  so  that  it  may 
always  assume  its  proper  position.     The  pivot  should  there- 
fore be  of  very  hard  steel  ground  to  a  fine  point.     In  the 
centre  of  the  needle  there  should  likewise  be  a  cup  of  agate 
or  some  other  hard  material  inserted  for  it  to  rest  upon. 

As  the  needle  is  generally  balanced  before  being  magnet- 
ized, the  north  end  in  northern  latitudes  will  always  "dip" 
after  the  magnetic  force  has  been  communicated  to  it.  To 
restore  the  balance,  a  coil  of  fine  brass  wire  is  wrapped 
around  the  south  end.  This  may  be  slipped  along  the  bar 
so  as  perfectly  to  restore  the  balance.  It  serves  also  to  dis- 
tinguish the  two  ends  of  the  needle. 

A  good  needle  will  vibrate  for  a  considerable  time  after 


SEC.  L]  DEFINITIONS   AND  INSTRUMENTS.  163 

having  been  disturbed.  If  it  settles  soon,  it  is  defective  in 
magnetic  power,  or  the  pivot  is  imperfect.  To  preserve  the 
pivot  in  good  order,  the  needle  should  always  be  lifted  from 
it  when  not  in  use. 

268.  The  Magnetic  Meridian,     The  line  upon  the  sur- 
face of  the  earth  in  the  direction  of  the  needle,  when  unin- 
fluenced by  disturbing  causes,  is  called  the  magnetic  me- 
ridian.    If  the  needle  pointed  steadily  to  the  north  pole, 
the  magnetic  meridian  would  coincide  with  the  true.     This 
is,  however,  far  from  being  the  case.     Throughout  the  east- 
ern part  of  the  United  States  and  Canada  it  points  west  of 
north,  the  amount  of  the  deviation  (called  the  variation  of  the 
compass)  being  different  in  different  places.     This  amount 
is  subject  to  a  gradual  secular  change.    (See  chap,  x.) 

269.  The  Magnetic  Bearing.    The  bearing  of  a  line 
from  the  magnetic  meridian  is  called  the  magnetic  bearing. 
This  has  generally  been  used  in  land  surveying.     Its  con- 
venience is  such  as  to  have  heretofore  counterbalanced  its 
defects  in  the  opinion  of  a  large  number  of  surveyors.     The 
attention  of  scientific  surveyors  and  legislators  has  of  late 
been  called  to  the  difficulties  arising  from  the  use  of  such  a 
false  and  varying  standard.     In  Pennsylvania,  by  a  late  law, 
the  bearings  of  all  lines  inserted  in  the  title-deeds  of  real 
estate  are  required  to  be  from  the  true  meridian  line.     The 
surveys  of  United  States  public  lands  have  always  been 
made  on  this  principle. 

270.  There  are  two  modes  in  which  the  needle  may  be 
employed  to  enable  us  to  determine  the  bearing  of  a  line. 

First.  Attached  to  the  needle  may  be  fixed  a  card  divided 
as  in  Fig.  114,  or  subdivided  into  degrees, — the  north  point 
of  the  needle  being  directly  under  the  north  point  of  the 
card.  Such  a  card  would  always  place  itself  in  the  same 
position  with  respect  to  the  cardinal  points. 

To  determine  the  bearing  of  a  line,  it  would  only  be 
necessary  to  have  a  pair  of  sights  in  the  line  of  a  diameter 
of  the  card,  with  an  index  between  them  to  show  at  what 


16£  COMPASS  SURVEYING.  [CHAP.  Y. 

point  of  the  card  the  line  crossed.  The  degrees  between 
this  point  and  the  north  or  south  point  of  the  card  would 
be  the  bearing  required.  Thus,  the  bearing  of  AB  would 
be  about  N".  67°  E.  The  cardinal  points  on  tfre  card  show 
the  points  between  which  the  line  runs. 

The  great  defect  in  this  plan  is  that,  in  consequence  of  the 
weight  of  the  card,  the  needle  settles  slowly,  and  the  pivot 
is  very  liable  to  wear.  The  card,  too,  must  be  made  of  some 
light  material,  which  cannot  be  divided  so  accurately  as 
metal.  This  form  is  therefore  never  used  except  for  the 
mariner's  compass. 

Second.  The  sights  may  be  connected  with  a  circular  box 
in  the  centre  of  which  is  the  pivot, — the  circumference  of 
the  box  being  appropriately  divided.  This  is  the  plan  em- 
ployed in  the  surveyor's  compass  or  circumferentor. 

271.  The  Compass.  The  compass  consists  of  a  stiff 
brass  plate  A,  (Figs.  115,  116,)  carrying  the  circular  box  B, 
and  furnished  at  the  ends  with  two  brass  sights  C,  perpen- 
dicular to  its  plane.  In  the  centre  of  the  box  is  the  pivot 
to  support  the  magnetic  needle. 

The  circumference  of  the  box  is  divided  into  360°,  and 
these  in  the  larger  instruments  are  subdivided  into  halves. 

The  zero-points  are  in  the  line  joining  the  sights,  one 
being  marked  for  the  north,  and  the  other  for  the  south. 
The  degrees  are  counted  from  zero  to  90°  each  way. 

If  we  stand  opposite  the  south  point  looking  towards  the 
north,  the  90°  on  the  left  hand  is  marked  E.  and  that  on 
the  right  W.  The  cardinal  points  thus  follow  each  other 
in  an  inverted  order. 

The  reason  why  this  should  be  so  will  appear  from  con- 
sidering the  difference  between  the  mariner's  compass  and 
the  circumferentor.  In  the  former,  the  card  is  stationary, 
while  the  index  moves;  in  the  latter,  the  index,  which  is  the 
needle,  is  stationary,  while  the  divided  circle  moves :  while, 
then,  the  north  point  of  the  box  is  moving  towards  the  east, 
the  north  point  of  the  needle  will  traverse  it  towards  the 
west.  In  order,  then,  that  the  index  should  not  only  point 
to  the  number  of  degrees,  but  also  show  the  cardinal  points 


SEC.  L] 


DEFINITIONS  AND  INSTRUMENTS 


165 


between  which  the  line  runs,  those  points  must  be  engraved 
in  a  reverse  order. 

Thus,  supposing  the  instrument  to  be  in  the  position,  (Fig. 
115,)  the  north  point  of  the  needle  at  L  shows  the  magnetic 


north,  and  the  south  point  the  magnetic  south;  the  point 
midway  between  these  to  the  right  is  east.  The  line  from  C 
to  C'  is  therefore  south  of  east.  If  then  the  north  point  of  the 
needle  is  to  be  used  as  the  index,  it  should  be  found  between 
the  letters  S.  and  E.  The  bearing  in  the  figure  is  S.  80°  E. 


166  COMPASS   SURVEYING.  [CHAP.  V. 


272.  The  Sights.     These  consist  of  two  plates  of  brass 
about  an  inch  wide  set  at  right  angles  to  the  plate.     Each 
plate  has  a  vertical  slit  cut  in  it,  with  larger  openings  at 
intervals,  as  seen  in  Fig.  116  at  H.     The  faces  of  the  sights 
are  seen  at  G.     The  slits  should  be  perfectly  straight,  and 
as  narrow  as  is  consistent  with  distinct  vision.     The  largei 
openings  enable  the  surveyor  to  see  the  object  more  readily 
than  he  could  through  the  fine  slits. 

Instead  of  the  sights,  a  telescope  that  can  be  elevated  or 
depressed  in  a  plane  perpendicular  to  that  of  the  plate  A  is 
sometimes  employed.  It  has  the  advantage  of  giving  more 
distinct  vision  at  great  distances,  and,  when  connected  with 
a  vertical  arc,  of  determining  the  angle  of  elevation  of  a  hill 
up  or  down  which  the  line  may  run.  This  object  may  be 
obtained  with  the  sights,  by  having  at  the  lower  end  of  one 
of  them  a  projection  pierced  with  a  small  hole,  and  upon 
the  face  of  the  other  the  angles  of  elevation  engraved.  By 
looking  through  the  hole  at  an  object  on  the  summit  of  the 
hill,  the  angle  of  elevation  may  be  read  on  the  face  of  the 
engraved  sight. 

If  such  a  scale  is  not  on  the  instrument,  it  may  be  put  on 
by  the  surveyor  himself;  a  mark  being  made  on  one  sight 
near  the  bottom,  or  a  small  plate  with  a  hole  being  screwed 
to  it ;  on  the  other,  at  the  same  distance  from  the  plate,  the 
zero  mark  should  be  made.  The  distance  from  zero  to  the 
other  marks  will  be  the  tangent  of  the  angle  of  elevation 
to  a  radius  equal  to  the  distance  between  the  sights. 
Measure  therefore  accurately  the  distance  between  the 
sights,  and  say,  As  rad.  :  tangent  of  the  number  of  degrees 
: :  the  distance  between  the  sights  :  the  distance  from  the 
zero  point  to  the  mark  for  that  number  of  degrees. 

273.  Attached  to  the  plate  there  are  generally  two  levels 
at  right  angles  to  each  other,  as  in  the  transit  and  theodolite. 

274.  The  Verniers.     In  some  instruments,  the  compass- 
box  is  movable   about  its   centre  for  a  few  degrees,  the 
amount  of  deflection  being  determined  by  the  vernier  Y. 
The  purpose  of  this  arrangement  will  appear  hereafter. 


SEC.  L]  DEFINITIONS  AND  INSTRUMENTS.  167 


Fig.  116. 


C' 


168  COMPASS   SURVEYING.  [CHAP.  V. 

275.  In  the  figures  115,  116,  the  different  parts  described 
above  are  lettered  as  below.     Different  makers,  however, 
arrange   the  parts  differently.     A  is  the  principal  plate, 
which  bears  all  the  other  parts.     B  is  the  compass-box, 
sometimes  movable  about  its  centre  by  means  of  a  pinion 
connected  with  the  milled  head  I,  and  capable  of  being 
clamped  in  any  position  by  the  screw  K.     D  is  the  needle, 
resting  on  a  pivot  in  the- middle  of  the  compass-box.     The 
needle  can  be  raised  from  its  pivot  by  the  screw  F.     C  and 
C'  are  the  sights,  which  are  fastened  to  the  plate  by  the 
screws  N.     M,  M  are  the  levels. 

276.  The  Pivot.     This  should,  as  remarked  above,  be 
extremely  hard  and  very   sharp.     It   should  likewise   be 
placed  exactly  in  the  centre  of  the  box  and  in  the  line  join- 
ing the  slits  in  the  sights. 

To  discover  whether  it  is  properly  centred,  and  likewise 
whether  the  needle  is  straight,  turn  the  compass  until  the 
north  point  of  the  needle  coincides  with  any  given  number 
of  degrees.  The  south  point  must  be  180°  distant.  If  it 
is  so  in  all  positions,  or,  in  four,  distant  90°,  as  for  instance 
the  O's  and  90's,  the  needle  is  straight  and  well  centred. 

Draw  a  hair  or  fine  silk  string  through  the  slits  in  the 
sights.  If  this  passes  over  the  zero-points,  the  centre  is  in  line. 

Or,  sight  to  a  very  near  object,  and  note  the  reading.  Turn 
the  instrument  half  round,  and  again  note  the  reading :  if 
these  do  not  agree,  the  pivot  is  not  on  the  line  of  sight. 
Half  the  difference  is  the  actual  error. 

277.  The  Divided  Circle.    The  accuracy  of  the  division 
may  be  tested  by  turning  the  plate  into  different  positions. 
If  in  all  cases  the  opposite  ends  of  the  needle  point  to  the 
same  number  of  degrees,  the  probability  is  that  the  circle  is 
correctly  divided. 

If  the  compass  has  a  vernier,  set  the  instrument  in  any 
direction.  Then  move  the  box  through  any  number  of 
degrees,  and  see  whether  the  needle  traverses  the  same 
number  of  degrees  as  the  vernier.  If  it  does  in  all  posi- 
tion?, the  arc  is  properly  divided. 


SEC.  I.]  DEFINITIONS  AND   INSTRUMENTS.  169 

278.  Adjustments.      The   levels  may  be  adjusted  as 
directed  for  the  transit  and  theodolite. 

The  sights  should  be  perpendicular  to  the  plane  of  the 
instrument.  To  verify  this,  suspend  a  long  plumb-line: 
level  the  plate,  and  sight  to  this  line.  If  it  appears  equally 
distinct  through  all  parts  of  the  slit,  the  sight  is  perpen- 
dicular. Turn  the  instrument  half  round  and  test  the  other 
sight  in  the  same  manner.  If  either  is  found  incorrect,  the 
maker  should  rectify  it. 

279.  The  compass,  as  already  remarked,  is  very  generally 
used  for  surveying  purposes,  though  it  is  fast  giving  place 
to  the  transit.     The  latter  is  furnished  with  a  compass-box, 
which  was  not  described  with  the  instrument,  as  it  was  not 
needed  at  that  stage  of  the  work.     It  is  in  all  respects 
similar  to  the  box  attached  to  the  compass  itself.     The 
theodolite  likewise  has  a  compass.     It  is,  however,  so  small 
as  to  be  of  very  little  use  in  accurate  work. 

280.  The  compass  k  generally  supported  on  an  axis  in- 
serted in  the  socket  0.     This  axis  terminates  in  a  ball, 
which  works  freely  but  firmly  in  a  socket.     This  arrange- 
ment admits  of  the   axis  being  placed  in  any  direction. 
The  compass-plate  may  thus  be  made  level. 

Instead  of  a  tripod,  many  surveyors  prefer  a  single  staff 
pointed  with  iron.  This  is  called  a  "Jacob's  Staff."  Its 
chief  defects  are  the  difficulty  of  setting  in  hard  ground  or 
among  stones,  and  the  want  of  steadiness  in  windy  weather. 

281.  Defects  of  the  Compass.     Though  a  very  con- 
venient and  useful  instrument,  the  compass  is  deficient  in 
two  very  important  particulars : — its  indications  are  neither 
correct  nor  precise. 

It  is  not  correct,  because,  as  already  remarked,  the  lieedle 
(which  is  the  standard)  does  not  do  what  it  professes :  it 
does  not  point  to  the  north.  This  would  be  of  compara- 
tively little  importance  if  its  direction  were  fixed  or  paral- 
lel; but  neither  of  these  is  the  fact.  It  not  only  varies 


170  COMPASS  SURVEYING.  [CHAP.  V. 

from  year  to  year,  but  from  season  to  season,  and  even 
during  the  same  day.  These  variations  will  be  the  subject 
of  a  future  chapter. 

The  presence  of  ferruginous  matter  in  the  earth,  or  the 
too  great  proximity  of  the  chain,  or  of  any  other  piece  of 
iron,  may  deflect  it  very  seriously  from  its  normal  position. 

It  is  not  precise.  The  divisions  on  the  arc  are  rarely 
smaller  than  half-degrees ;  and  if  they  were  finer  it  would 
be  difficult  to  read  to  less  than  a  quarter  of  a  degree.  A 
little  calculation  will  convince  one  that  this  is  a  serious 
defect  where  accuracy  is  desired.  An  error  of  5'  in  the 
bearing  would  cause  a  deviation  of  nearly  one  foot  in  ten 
chains,  or  about  seven  feet  eight  inches  in  a  mile. 


SECTION  II. 
FIELD  OPERATIONS. 

282.  Bearings.  To  take  the  bearing  of  a  line,  set  the 
compass  directly  over  one  end ;  level  it,  and  turn  the  plate 
till  the  other  end  of  the  line — or  a  rod  set  up  in  the  direc- 
tion of  the  line  at  a  distance  as  great  as  is  consistent  with 
distinct  vision — can  be  seen  through  the  slits.  Then,  when 
the  needle  has  settled,  notice  the  number  of  degrees  to 
which  the  end  of  the  needle  points,  and  the  cardinal  points 
between  which  it  is  situated :  the  result  will  be  the  bearing 
of  the  line. 

If  the  north  end  of  the  compass  is  ahead,  the  north  end 
of  the  needle  should  be  used,  and  vice  versa. 

If  you  are  running  with  the  north  end  of  the  compass 
ahead,  and  the  north  point  of  the  needle  is  between  S.  and 
E.  and  points  to  45|°,  the  bearing  is  S.  45J°  E. 

In  reading,  the  eye  should  be  placed  opposite  to  the  other 


SEC.  II.]  FIELD  OPERATIONS.  171 

end  of  the  needle ;  otherwise,  owing  to  the  parallax  of  the 
point,  it  will  appear  to  stand  at  a  different  point  of  the  arc 
from  whatsit  really  does.  Any  iron  about  the  person  will 
be  less  likely  to  affect  the  needle  than  when  in  another 
position. 

283.  Use  of  the  Vernier.    When  the  needle  does  not 
point  to  one  of  the  divisions  of  the  arc,  it  is  usual  to  esti- 
mate the  fraction.       Some  surveyors,  however,  after  the 
needle  has  come  to  rest,  notice  between  which  divisions  the 
needle  points,  and  then  move  the  compass-box,  by  turning 
the   milled  head  I,  until  the  point  of  the  needle  is  op- 
posite one  of  the  divisions.      The  amount  by  which  the 
box  is  turned,  as  indicated  by  the  vernier,  will  give  the 
fraction. 

This  plan,  though  theoretically  correct,  adds  really 
nothing  to  the  correctness  of  the  work.  The  liability  to 
derangement,  from  handling  the  instrument,  is  so  great  as 
to  neutralize  any  advantage  it  might  otherwise  possess. 

284.  Reverse  Bearing.     The  reverse  bearing  of  every 
line  should  be  taken.     To  do  this,  set  the  compass  at  the 
position  of  the  rod,  and  sight  back  to  the  former  station. 
The  bearing  found  should  be  the  reverse  of  the  former.     If 
it  is  not,  the  work  at  the  former  station  should  be  reviewed ; 
if  found  correct,  the  difference  between  the  two  must  arise 
from  some  local  cause. 

285.  Local  Attraction.    When  the  back  sight  does  not 
agree  with   the   forward   sight,   some  cause   of  derange- 
ment exists  about  one  of  the   stations.      This  is   called 
local  attraction.      It  is  generally   caused  by  ferruginous 
matter  in   the   earth.      It  is   said  that  any  high   object, 
such  as  a  building  or  even  a  tree,  will  slightly  deflect  the 
needle.      In  situations  in  which  trap  rocks   abound,  the 
local  attraction  is  often  very  great.     The  author  has  known 
a  variation  of  more  than  10°  in  a  line  of  two  and  a  half 
chains  long,  produced  by  this  cause  alone.    In  such  regions, 
running  by  the  needle  is  very  troublesome,  and  may  cause 


172  COMPASS  SURVEYING.  [CHAP.  V. 

very  serious  errors  unless  great  care  is  taken  to  allow  for 
the  effect  produced. 

To  discover  where  the  attraction  exists,  select  a  number 
of  positions  in  the  neighborhood  of  the  suspected  points, 
and  note  their  bearings  from  these  stations,  and  also  from 
each  other.  The  agreement  of  several  of  these  will  prove 
their  probable  correctness.  The  points  thus  found  to  be 
void  of  local  attraction  may  be  taken  as  the  starting 
points. 

In  surveying  a  farm,  a  very  good  way  is  to  note  the 
forward  and  back  sights  of  every  line.  If  these  are  found 
to  agree  on  any  line,  they  may  be  presumed  to  be  right,  and 
the  others  corrected  accordingly. 

286.  To  correct  for  back  sights. 

"When  the  back  sight  is  greater  than  the  fore  sight,  sub- 
tract the  difference  from  the  next  bearing,  if  the  two  lie 
between  the  same  points  of  the  compass  or  between  points 
directly  opposite,  but  add  it  in  all  other  cases.  If  the  back 
sight  is  the  less,  add  the  difference  in  the  former  case,  and 
subtract  it  in  the  latter. 

Where  the  local  attraction  is  great,  or  the  -line  runs 
nearly  in  the  direction  of  one  of  the  cardinal  points,  a  diffi- 
culty may  occur  in  the  application  of  the  preceding  rule. 
A  little  reflection  will  enable  the  surveyor  to  modify  it  to 
suit  the  case. 

287.  By  the  Vernier.     It  is  more  convenient  in  practice 
to  turn  the  box  by  the  vernier  until  the  reading  for  the 
back  sight  corresponds  with  the   fore  sight.     The   needle 
will  then  give  the  true  bearing  of  the  new  line  as  though 
no  attraction  existed. 

288.  To    survey  a   Farm.       Commence    by   going 
round  it,  and  verifying,  so  far  as  can  be  done,  the  land- 
marks, fixing  stakes  at  the  corners,  so  that  the  assistant 
may  readily  find  them  if  he  is  not  already  familiar  with 
their  position.     Then,  placing  the  compass  at  one  corner, 


SEO.  II.]  FIELD  OPERATIONS.  173 

send  the  flag-man  ahead  to  the  next  corner ;  note  the  bearing 
of  his  pole ;  and  so  proceed  with  the  sides,  in  succession, 
taking  a  back  sight  at  each  station. 

If  the  end  of  the  line  cannot  be  seen  from  the  begin- 
ning, let  the  flag-man  erect  his  pole,  in  the  line,  at  a  point 
as  distant  from  the  beginning  as  possible.  Sight  to  the 
pole,  as  before;  then,  going  forward,  set  the  compass  by 
sighting  to  the  last  station.  The  flag-man  should  now  be 
placed,  exactly  in  line,  at  another  station.  So  proceed 
until  the  end  of  the  line  has  been  reached. 

289.  Random  Line.  If  the  first  position'  of  the  flag- 
staff were  not  exactly  in  line,  the  course  run  will  deviate 
to  the  right  or  left  of  the  corner.  "Where  such  is  the  case, 
measure  the  perpendicular  distance  to  the  corner,  and  de- 
termine the  correction  by  the  following  rule : — 

As  the  length  of  the  line  is  to  the  deviation  found  as 
above,  so  is  57.3  degrees,  or  3438  minutes,  to  the  correction 
in  the  bearing.* 

In  running  through  woods,  it  is  very  frequently  necessary 
to  correct  the  bearing  in  this  manner.  In  all  cases,  how- 
ever, where  back  sights  are  taken,  the  compass  should  be 
allowed  to  stand  at  the  last  station  on  the  random  line, 
since  the  local  attraction  often  varies  very  considerably 
in  a  short  distance.  If  it  is  desired  to  run  the  next  line 
precisely  on  its  location,  the  corner  should  be  sighted  to 
from  the  end  of  the  random  line,  and  a  back  sight 
taken. 


*  This  rule  is  founded  on  the  ordinary  rule  for  the  solution  of  right-angled 
triangles, — the  length  being  the  hypothenuse,  and  the  deviation  the  perpen- 
dicular, an  arc  of  57.3  degrees  being  equal  in  length  to  the  radius. 

Thus,  supposing,  in  running  a  line  N.  85°  30X  E.  27.53  chains,  the  corner  is 
found  35  links  to  the  right  hand  :  the  calculation  would  be 

27.53  :  35  : :  67.3°  :  0°  43'. 
The  proper  bearing  would  therefore  be  N.  36°  13'  E. 


174  COMPASS  SURVEYING.  [CHAP.  V. 

290.  When  the  far  end  of  the  line  cannot  be  seen,  it 
will  sometimes  be  found  convenient  to  run  to  a  station  as 
near  the  middle  of  the  line  as  possible,  if  one  can  be  found 
from  which  both  ends  can  be  seen.     Then,  instead  of  con- 
tinuing on  in  the  same  course,  sight  to  the  corner.     The 
chain-men  should  note  the  distance  to  the  assumed  station. 
A  very  obtuse-angled  triangle  will  thus  be  formed,  and  the 
correction  in  bearing  may  be  readily  calculated. 

Thus,  supposing  the  line  were  AB,  (Fig.  117,) 
passing  over  an  elevation  at  C.  At  A  the  bearing 
of  AC  was  found  to  be  K  43|°  W.,  distance 
10.50  chains.  At  C,  CB  was  K  43°  W.,  distance 
7.36  chains. 

We  have  AC  :  BC  ::  sin.  B  :  sin.  A; 

or,  as  the  angles  are  small,  AC  :  BC  ::  B  :  A; 
whence  AC  +  BC  :  BC  : :  B+A  :  A. 

That  is,  17.86  :  7.36  : :  45'  :  A  =  19',  the  required 
correction.  The  true  bearing  of  AB  is  therefore 
K  43J°  W. 

Where  the  deviation  from  the  correct  line  is  not  much 
greater  than  in  the  example  given,  AB  is  sensibly  equal  to 
AC  -f-  CB.  Where  the  deviation  is  considerable,  the  angles 
and  side  should  be  calculated  by  Trigonometry. 

The  above  rule  may  be  expressed  thus : — 

As  the  sum  of  the  distances  is  to  the  last  distance,  so  is 
the  whole  deviation  to  the  correction  to  be  applied  at  the 
first  station. 

291.  Proof  Bearings.      In  the  course  of  the  survey, 

bearings  or  angles  should  be  taken  to  prominent  objects. 
These  form  a  test  of  the  accuracy  of  the  work.  Three 
bearings  are  necessary  to  each  object:  two  of  these,  being 
required  to  fix  its  position,  will  afford  no  check  on  the  inter- 
mediate measurements ;  but  their  coincidence  with  a  third 
will  determine  the  probable  correctness  of  all,  and  of  the 
connecting  measurements.  Diagonal  bearings  and  dis- 
tances may  likewise  be  taken  as  proof  lines. 


SEC.  II.]  FIELD  OPERATIONS.  175 

292.  Angles  of  Deflection.  In  surveying  with  the 
transit  or  theodolite,  it  is  most  convenient  to  record  the 
angles  of  deflection;  that  is,  the  angle  by  which  the  new 
course  deviates  to  the  right  or  to  the  left  from  that  of  the 
last  line.  This  is  always  done  in  surveying  roads,  rivers,  &c. 
From  the  angles  of  deflection  the  hearings  are  very  readily 
deduced,  by  rules  to  be  given  hereafter.  As  checks  to  the 
work,  the  bearings  of  some  of  the  lines  may  likewise  be 
taken. 

In  a  closed  survey  the  whole  deflection  must  equal  360°. 
To  determine  whether  it  is  so,  arrange  the  deflections  to 
the  left  in  one  column,  and  those  to  the  right  in  another. 
Sum  the  numbers  in  each  column :  the  difference  of  these 
sums  should  equal  360°. 

In  practice  this  will  rarely  occur;  though  in  open  ground, 
where  the  angles  can  readily  be  taken,  the  error  should  not 
exceed  four  or  five  minutes  in  a  tract  of  ten  or  twelve  sides, 
provided  a  good  transit  or  theodolite  is  employed. 

EXAMPLE. 

The  following  are  the  notes  of  a  survey  taken  by  the 
author:— 1.  S.  53°  10'  W.;  2.  Deflect  97°  3'  to  the  right; 
3.  97°  45'  to  the  right;  4.  81°  14'  to  the  right;  5.  30° 
12'  to  the  left;  6.  12°  14'  to  the  left;  7.  27°  48'  to  the 
right.  Whence  the  first  line  deflects  98°  34'  to  the  right. 

Right  hand.  Left  hand. 

97°    3'  30°  12' 

97°  45'  12°  14' 

81°  14'  42°  26' 

27°  48' 
98°  34' 


402°  24' 

42°  26' 
359°  58', 

differing  but  two  minutes  from  360°. 


176  COMPASS  SURVEYING.  [CHAP.  V. 

Where  the  difference  amounts  to  several  minutes,  it  is 
best  to  distribute  it  among  the  angles. 

The  rule  which  is  sometimes  given:  to  determine  the 
angles  from  the  bearings,  and  ascertain  whether  the  sum 
of  the  internal  angles  is  equal  to  twice  as  many  right  angles 
as  the  figure  has  sides,  less  four  right  angles — proves  nothing 
in  regard  to  the  correctness  of  the  field  work.  Any  set  of 
bearings  will  prove  in  this  way. 


SECTION  III. 

OBSTACLES  IN  COMPASS  SURVEYING,* 

A.— PROBLEMS  IN  RUNNING  LINES. 

293.  MANY  of  the  obstacles  that  occur  in  angular  sur- 
veying have   already   been    alluded   to.      These,   and  all 
others  which  the  operator  will  meet  with,  may  be  over- 
come  by  the  principles  of  Trigonometry.     As,  however, 
there  is  frequently  a  choice  in  the  means  to  be  used,  the 
following  methods  are  given,  as  being  perhaps  the  most 
simple : — 

294.  Problem  1. —  To  run  a  line  making  a  given  angle  with 
a  given  line  from  a  given  point  within  it. 

Place  the  instrument  at  the  point,  and  sight  along  the 
line.  Tarn  the  plate  the  required  number  of  degrees,  and 
the  sights  or  telescope  will  be  in  the  required  line. 

*  Mo.i:y  more  such  methods  may  be  found  in  Gillespie's  "Land  Surveying." 


SEC.  III.] 


OBSTACLES  IN  COMPASS  SURVEYING. 


177 


295.  Problem  2. — To  run  a  line  making  a  given  angle  with 
a  given  inaccessible  line  at  a  given  point  in  that  line. 

Let  AB  (Fig.  118)  be  the  given  Kg-  "8. 

line,  and  A  the  given  point. 
Take  two  points  C  and  D  from 
which  A  and  some  other  point  B 
in  AB  may  be  seen,  and  measure 
CD.  Then  take  the  angles  ACD, 
BCD,  ADC,  and  BDC.  The  dis- 
tance AC  and  the  angle  CAB 
may  be  calculated. 

Run  CE,  making  ACE  =  CAB :  CE  will  then  be  parallel 
to  AB.  Now,  if  we  suppose  AE  to  be  drawn,  we  shall 
have  in  the  triangle  ACE  all  the  angles  and  side  AC  to 
find  CE.  Lay  off  this  distance  from  C  to  E,  and  run  the 
line  EF  towards  A. 

If  A  cannot  be  seen  from  E,  calculate  CEF,  and  run  the 
line  from  E,  making  the  proper  angle  with  CE. 


Problem  3. — From  a  given  point  out  of  a  line,  to  run  a 
line  making  a  given  angle  with  that  line. 

296.  Where  the  line  is  accessible. 

If  the  compass  is  used.  Take  the  bearing  of  the  given 
line.  Then  place  the  compass  at  the  given  point,  and 
set  it  to  same  bearing.  Deflect  the  compass  the  number  of 
degrees  required,  and  run  the  line. 


Fig.  119. 


B' 


If  a  transit  or  theodolite  is  used. 
Set  the  instrument  at  some  point  ^ 
A  (Fig.  119)  in  the  line,  and  take 
the  angle  BAG.  Move  the  instru- 
ment  to  C,  and  make  the  angle 
ACB  =  B  -  A,  or  =  180°  -  (B  + 
A),  and  CB  or  CB'  will  be  the  line  required. 

In  all  cases,  unless  the  line  is  to  be  a  perpendicular,  there 
will  be  two  lines  that  will  answer  the  conditions. 

12 


178 


COMPASS  SURVEYING. 


[CHAP.  V. 


Fig.  120. 

F  B 


297.  If  the  line  is  inaccessible.     Let 
AE  (Fig.  120)  be  the  given  line,  and 
C  the  given  point.      Run  any  con- 
venient base  CD,  and  take  the  angles 
of  position  of  two  visible  points  A 
and  B  in  the  given  line.     Then,  in  the 
triangle  ADC,  we  shall  have  DC  and 

the  angles,  to  find  CA.  Similarly,  in  CBD,  find  CB. 
Then,  in  ACB,  we  shall  have  AC,  CB,  and  ACB  to  find 
ABC. 

Run  CF,  making  BCF  =  B  -  F,  or  180°  -  (B  +F),  and 
it  will  make  the  required  angle  with  AB. 

298.  If  the  point  be  inaccessible. 
From   any  convenient  stations  A 
and  B  (Fig.  121)  in  the  line  AB, 
take  the  angles  of  position  of  the 
point  C,  and  measure  AB.     Then, 
in    the    triangle  ABC,   we    shall 
have  the  angles  and  the  side  AB 
to  find  BC. 

In  BCD  we  then  have  the  angles  and  side  BC  to  find 
BD. 

BD  may  be  found  by  a  single  proportion,  thus : — 

Sin.  ACB .  sin.  BDC  :  sin.  BAG .  sin.  BCD  : :  AB  :  BD. 

For  we  have     sin.  ACB  :  sin.  BAG  : :  AB  :  BC, 
and  sin.  BDC  :  sin.  BCD  : :  BC  :  BD. 

Whence  (23.6) 
sin.  ACB  .  sin.  BDC  :  sin.  BAG  .  sin.  BCD  : :  AB  :  BD. 

Having  found  BD,  DC  may  be  run  towards  C ;  or  by  the 
angle,  if  C  be  invisible  from  D. 

If  C  is  visible  from  the  point  D,  the  latter  may  be  found 
by  trial,  thus : — 

Set  the  instrument  at  a  station  as  near  the  proper  posi- 
tion as  possible,  and  deflect  the  given  angle.  Notice 
whether  the  line  passes  to  the  right  or  left  of  the  point,  and 


SBC.  III.] 


OBSTACLES  IN  COMPASS  SURVEYING. 


179 


move  the  instrument  accordingly.     A  few  trials  will  put  it 
in  its  proper  place. 


Fig.  122. 
G 


299.  If  the  point  and  the  line  both 
be  inaccessible.  Take  any  convenient 
station  D,  (Fig.  122,)  and  run  DE 
parallel  to  AB,  by  Art.  302.  Then 
run  CFG,  making  the  required  angle 
with  ED,  by  Art.  298;  or  the  dis- 
tance on  the  base  DC  (Fig.  125) 
may  be  calculated. 


Problem  4. — To  run  a  line  parallel  to  a  given  line  through  a 
given  point. 

300.  If  the  line  be  accessible. 


Fig.  123. 


With  the  compass.    Take  the  bearing  of  the  given  line,  and 
through  the  given  point  run  a  line  with 
the  same  bearing. 

'With  the  transit  or  theodolite.  At  any 
point  A  (Fig.  123)  in  the  given  line 
take  the  angle  BAG.  Eemove  the 
instrument  to  C,  and  make  ACD  = 
BAC.  CD  will  be  parallel  to  AB. 

301.  If  the  point  be  inaccessible. 
At  A  and  B,  (Fig.  124,)  any  two 
points  in  the  given  line,  take  the 
angles  BAC  and  ABC.  Measure 
AB,  and  calculate  AC.  Make  CBD 
=  ACBandBD=AC.  Through 
D  run  DE  in  the  line  CD :  it  will  be  the  parallel  required. 


302.  If  the  line  be  inaccessible. 
From  C  (Fig.  125)  run  any  base- 
line CD ;  and  at  C  and  D  take 
the  angles  of  position  of  two 
visible  points  A  and  B  in  the 
given  line.  Calculate  the  angle 


Fig.  125. 


180 


COMPASS  SURVEYING. 


[CHAP.  V. 


CAB.    Bun  EOF,  making  ACE  =  CAB,  and  EF  is  the 
parallel  required. 


If  the  line  and  the  point  both  be  inaccessible. 


Fig.  126. 


Fig.  127. 


303,  First  Method. — Assume 
any  station  D,  (Fig  126,)  and 
run  a  line  DE  parallel  to  AB, 
by  Art.  302,  and  towards  C  run 
Fa  parallel  to  DE,  by  Art.  301. 


304.  Second  Method. — 
Take  any  convenient  base 
DE,  (Fig.  127,)  and  take 
the  angles  of  position  of 
C,  A,  and  B  at  D  and  E. 
Calculate  BE,  CE,  and 
EBA.  Then  CFB  =  180° 
—  EBA.  In  CEF,  we 
then  have  the  angles  and 
CE  to  find  EF.  Lay  off  EF  the  calculated  distance,  and  run 
the  line  from  F  to  C. 


B.— PROBLEMS  FOB  THE  PROLONGATION  AND  INTER- 
POLATION OF  LINES. 

305.  In  running  a  line,  obstacles  are  often  met  with 
which  it  requires  some  ingenuity  to  overcome,  and  which 
will  perplex  the  surveyor  unless  he  has  prepared  himself 
by  previous  study  of  all  cases  which  are  likely  to  occur. 
If  the  total  length  of  a  line  were  all  that  it  was  necessary 
to  determine,  the  two  points  at  its  extremity  might  be  con- 
nected by  a  series  of  triangles,  and  that  length  calculated 
by  Trigonometry;  but  it  is  generally  desirable  to  have  the 
line  marked  out  so  that  the  exact  position  of  the  dividing 
fence,  if  one  is  placed,  or  of  the  division  if  there  be  no 
fence,  may  be  indicated  by  stakes  or  by  marked  trees. 
To  do  this,  the  line  itself  must  be  traced,  or  another  run 


SEC.  III.] 


OBSTACLES  IN  COMPASS  SURVEYING. 


181 


in  its  neighborhood,  so  related  to  that  in  question  that  the 
surveyor  can  at  any  time  pass  from  the  one  to  the  other  to 
set  his  landmarks.  "We  shall  treat  of  the  different  kinds 
of  obstructions  likely  to  occur;  and,  as  the  prolongation 
and  interpolation  of  the  lines  are  generally  closely  con- 
nected with  the  determination  of  their  lengths,  the  two  will 
be  considered  together. 

Problem  1. — To  prolong  a  line  beyond  a  building  or  other 
obstruction. 

* 

306.  First  Method. — At  a  point  of  the  line  erect  a  per- 
pendicular of  such  length  as  to  pass  beyond  the  obstacle. 
Through  the  extremity  of  this  run  a  parallel  to  the  given 
line :  after  passing  the  obstacle,  pass  back  to  the  required 
line  by  an  equal  perpendicular.  The  distance  will  be  equal 
to  that  of  the  parallel. 


307.  Second   Method.— At  B 
(Fig.  128)  deflect  60°,  and  mea- 
sure EC.     At  C  deflect  120°, 
and  measure  CD  =  BC.    Deflect 
60°,  and  run  DE,  which  will  be 
in  line  with  AB.   BD=BC;for 
BDC  is  an  equilateral  triangle. 

308.  Third  Method.— At 
B  (Fig.  129)  deflect   60°, 
and  measure  BC.     At  C 
deflect  90°,  and  measure 
CD  =  1.732  times  BC.    At 
D  deflect  30°,  and  DE  will 
be  in  line  with  AB.    BD  = 


Fig.  128. 


Fig.  129. 


2BC. 


309.  Fourth  Method.— At 
B  (Fig.  130)  deflect  45°. 
Measure  BC.  At  C  turn 
90°,  and  make  CD  =  BC. 


At  D  turn  45°,  and  DE  will  be  in  line. 


Fig.  130. 


1.414  BC. 


182  COMPASS   SURVEYING.  [CHAP.  V. 

Problem  2. — To  interpolate  points  in  a  line. 

310.  If  one  end  be  visible  from  the  other.     Set  the  instru- 
ment at  one  end  and  sight  to  the  other:  an  assistant  can 
then  be   signalled  to  place   stakes   directly  in  line.      In 
crossing  a  valley,  determine   a   station,  as  above,  on  the 
borders,  from  which  the  valley  can  be  seen;   and,  placing 
the  instrument  at  this  point,  sight  to  a  similarly  deter- 
mined station  on  the  other  side.      Stations  may  thus  be 
determined  down  a  very  considerable  declivity.     With  the 
transit  almost  any  slope  may  be  sighted  down.     In  this 
operation,  the  instrument  must  be  very  carefully  levelled 
sideways ;   otherwise,  the  points  determined  in  the  valley 
will  be  out  of  line. 

311.  By  a  Random  line.  If  a  wood,  or  other  ob-    Fis- 131- 
struction,  prevents  one  end  of  the  line,  as  B,  (Fig. 

131,)  from  being  seen,  run  a  line  AC  as  nearly  in 
the  given  course  as  possible,  and  drive  a  stake  every 
five  or  ten  chains,  or  oftener  if  desirable.  When 
you  have  arrived  opposite  the  end  of  the  line,  note 
the  distance.  Also  measure  the  distance  CB  to 
the  end.  The  correction  of  the  bearing  may  be 
found  as  in  Art.  289,  and  the  points  be  inter-  / 
polated  as  in  Art.  209.  c  • 

312.  If  the  line  cannot  be  run  from  the  first  station. 

Lay  off  AC  (Fig.  132)  as  nearly  perpendicular    Fig.^ 
to  the  line  as  possible,  and  run  the  random  line 
CD.     On  arriving  opposite  the  end,  measure  DB. 
Then  say, — 

As  CD  is  to  the  difference  between  BD  and  AC, 
so  is  57.3°,  or  3438',  to  the  correction  of 
bearing. 

To  interpolate  points — Say,  as  CD  is  to  the 
distance  Ca  to  any  station  on  the  random  line,  so     ; 
is  the  difference  between  BD  and  AC  to  a  fourth  D[ 


\ 

\ 

-\b 

\ 

\ 


SEC.  III.]  OBSTACLES  IN  COMPASS    SURVEYING.  183 

term.  This  fourth  term' added  to  AC  if  BD  is  greater  than 
AC.,  but  subtracted  if  it  be  less,  will  give  the  correction  for 
the  point  a. 

If  the  random  line  crosses  the  other,  as  in  Fig.     Fis-  ^33- 
133,  say,  As  CD  is  to  the  sum  of  AC  and  BD,  so 
is  57.3°,     or    3438',    to    the    correction  of  the 
bearing. 

Points  may  be  interpolated  by  the  following 
rule : — 

Say,  As  CD  is  to  the  sum  of  AC  and  BD,  so 
is  the  distance  Ca  to  any  point  in  the  random  line 
to  a  fourth  term.  Take  the  difference  between 
this  fourth  term  and  AC. 

Then  if  AC  is  the  greater  of  the  two,  lay  off 
the  difference  on  the  same  side  of  the  random 
line  that  A  is ;  but  if  AC  be  the  less,  lay  off  the  remainder 
on  the  opposite  side. 

Where  a  point  in  the  line  at  a  given  distance  from  the 
beginning  is  required,  measure  that  distance  on  the  ran- 
dom line,  and  determine  the  offset  as  above. 

If  the  random  line  comes  out  very  distant  from  the  far 
station,  it  is  better  to  run  another  than  to  depend  on  that 
as  a  basis  for  interpolation. 

€.— PROBLEMS   FOR  THE  MEASUREMENT   OF  INAC- 
CESSIBLE DISTANCES. 

313.  The  various  methods  of  determining  the  lengths  of 
inaccessible  lines  are  merely  applications  of  the  rules  of 
Trigonometry,  and  might,  therefore,  be  applied  by  the  stu- 
dent without  further  instruction.  There  is,  however, 
always  a  choice  in  the  method  to  be  employed:  the  fol 
lowing  are  therefore  given,  that  all  that  is  needful  in  the 
case  may  be  brought  together. 

Problem  1. — To  determine  the  distance  between  two  points 
which  are  accessible  and  visible  from  each  other. 


184 


COMPASS   SURVEYING. 


[CHAP.  V. 


Fig.  134. 


314.  First  Method. — Select  any 
station  C,  (Fig.  134.)    Measure  BC, 
and  take    the    angles   BAG  and    B 
ABC.  Thence  we  can  calculate  AB. 

315.  Second  Method.  —  Measure 
CA  and  CB   (Fig.  134)  and    the 
angle  ACB;    whence,  having  two 
sides  and  the  included  angle,  AB 
may  be  determined. 

316.  Third  Method.— Where  the 
angles    can  be   taken    to   the    ex- 
tremities   of    an    inaccessible    but 
known  base  CD,  (Fig.  135,)  the  dis- 
tance    AB     may     be     calculated 
thus : — 


In  ABD  we  have      AD  :  AB  : :  sin.  ABD  :  sin.  ADB, 
and  in  ABC  we  have  AB  :  AC  : :  sin.  ACB  :  sin.  ABC. 

Whence  (23.6)  AD  :  AC  : :  sin.  ABD .  sin .  ACB  :  sin.  ADB  . 
sin.  ABC. 

Then,  in  CAD  having  the  ratio  of  AC  to  AD  and  the 
angle  CAD,  we  may  find  the  other  angles  by  Art.  141, 
thus : — 

As  AD  :  AC,  or  sin.  ABD  .  sin.  ACB  :  sin.  ADB  .  sin. 
ABC  : :  r  :  tan.  x,  and  as  rad.  :  tan.  (£«^45°)  : :  tan.  J  (ACD 
+  ADC)  :  tan.  J  (ACD  «*  ADC.) 

Having  now  the  angles  and  one  side  of  ACD,  AD  is 
found ;  whence,  in  ADB,  AB  may  be  determined. 

Thus,  sin.  CAD  :  sin.  ACD  : :  CD  :  AD, 

and  sin.  ABD  :  sin.  ADB  : :  AD  :  AB. 

Whence  (23.6)  sin.  CAD  .  sin.  ABD  :  sin.  ACD .  sin.  ADB 
• :  CD  :  AB. 


SEC-  III.]  OBSTACLES  IN  COMPASS  SURVEYING.  185 

EXAMPLES. 

To  determine  the  distance  AB,  accessible  at  its  extremi- 
ties, I  took  the  angles  to  the  ends  of  a  line  CD  10.75 
chains  long,  as  follows:— BAG  =  100°  35';  BAD,  48°  19'; 
ABC,  46°  15';  and  ABD,  85°  23'.  Bequfred  the  distance 
AB. 

ACB  =  180°  -  (BAG  +  ABC)  =  33°  10'. 
ADB  =  180°  -  (BAD  +  ABD)  =  46°  18'. 


r  sin.  ABD     85°  23'     A. 

C.  0.001411 

^  OY  \  sin.  ACB      33°  10'      " 

"   0.261952 

r  sin.  ADB     46°  18' 
ACor\sin.  ABC     46°  15' 

9.859119 
9.858756 

rad. 

10.000000 

tan.  x                          43°  45'  46" 

9.981238 

45 

tan.    45°  -  x           1°  14'  14" 

8.334392 

ACD  +  ADC 

10  3092^)8 

ACD  -ADC 

J-Vf»OV/£/^yt^O 

8  6436^0 

O»\J^tO\J(J\J 

ACD           66°  23'  14" 

f  sin.  CAD        52°  16'     A. 

C.  0.101896 

B  \sin.ABD  85°  23'     "  "   0.001411 

.    j  sin.  ACD  66°  23' 14"  9.962025 

\  sin.  ADB  46°  18'  9.859119 

: :  CD  10.75  ch.  1.031408 

:    AB  9.034  ch.  0.955859 

Problem  2. — To  determine  the  distance  on  a  line  to  the  in- 
accessible but  visible  extremity. 

' 

317.  This  may  be  done  by  the  methods  explained  in 
Arts.  236,  237,  and  238,  using  the  transit  or  theodolite  in 
running  the  lines,  or  by  the  following  method : — 

318.  Bun  a  base  line  from  a  point  in  the  line  making  any 


186 


COMPASS  SURVEYING. 


[CHAP.  V. 


angle  therewith,  and  at  its  extremity  take  the  angle  of  posi- 
tion of  the  point.  A  triangle  is  thus  formed  of  which  the 
angles  and  one  side  are  known. 

In  this  operation  the  triangle  should  be  made  as  nearly 
equilateral  as  possible. 

Problem  3. — To  determine  the  distance  when  the  end  is  in- 
visible and  inaccessible. 


319.    First   Method.— De-  Fis-136- 

fleet  at  B  (Fig.  136)  by  any 
angle,  and  measure  BD  to  a 
point  from  which  C  is  visi- 
ble. TakeBDC.  Then  calcu- 
late BC.  The  angle  C  should 
be  made  as  large  as  possible. 

If  AB  will  not  certainly 
pass  through  C,  operate  by  the  second  method. 


Fig.  137. 


-vo 


320.  Second  Method. — Bun 
EBD  making  any  angle  with 
AB,  (Fig.  137.)  Take  the  angles 
D  and  E.  In  DEO  find  DC. 
Then  in  DCB  we  have  two  sides 
DC  and  DB  and  the  included 
angle  to  find  BC  and  DBC.  If 
DBC  is  equal  to  ABE,  C  is  in 
AB  produced. 


Problem  4. — To  determine  the  distance  to  the  intersection  of 
two  inaccessible  lines. 


SEC.  III.] 


OBSTACLES   IN  COMPASS   SURVEYING. 


187 


321.     Let  AB  and 

CD  (Fig.  138)  be  the 
lines,  their  intersec- 
tion E  being  both  in- 
visible and  inaccessi- 
ble. It  is  required  to 
run  a  line  from  a 
given  point  G,  that 
shall  pass  through  E, 
and  to  determine  GE. 

Run    any   base  line 
GH,  and  take  the  angles  of  position  of  the  points  A,  B,  C, 
and  D  on  the  given  lines. 

Find  GO,  GD,  and  GDC ;  also  GA,  GB,  and  GBA.  Then, 
in  GBD,  we  have  GB,  GD,  and  BGD,  to  find  GBD,  GDB, 
and  BD.  In  BDE  we  then  have  BD  and  the  angles  to  find 
BE.  Finally,  in  GBE  we  have  GB,  BE,  and  the  included 
angle,  to  find  BGE  and  GE.» 


If  the  lines  AB  and  CD 
were  accessible,  the  line  GE 
might  be  run  by  Art.  212, 
and  the  distance  determined 
by  taking  the  angles  C  and 
G,  (Fig.  139.) 


Fig.  139f 


sm.  GEC 


Problem  5. — To  determine  the  distance  between  two  inac- 
cessible points. 


Fig.  140. 
B 


322.  First  Method.— Select  if       A 

possible  a  point  C,  in  the  direc-   rr 

tion  of  the  line  AB,  (Fig.  140.)  %%Xv 

From  a  station  D,  take  ADB  *X% 

and  BDC,   and   measure  DC. 
Then    in    CDB  we  have   CD 
and  the  angles  to  find  CB,  and 
in  CDA  we  have  CD  and  the  angles  to  find  CA. 
AB  =  CA  -  CB. 


188  COMPASS   SURVEYING.  [CHAP.  Y. 

323.  Second  Method. — Take  a  base  line  CD,  (Fig.  135,) 
which,  if  possible,  should  be  chosen  nearly  parallel  to  AB, 
and  not  much  shorter  than  it.     From  C  and  D  take  the 
angles  of  position  of  A  and  B,  whence  AB  may  be  calcu- 
lated. 

324.  Third  Method. — If  no   two   points  can  be  found 
whence  A  and  B  can  both  be  seen,  the  distance  can  be  found 
as  in  Prob.  9,  p.  114. 

325.  Fourth  Method. — If  A  and  B  can  both  be  seen  from 
no  one  station,  the  distance  may  be  found  by  Prob.  13, 
p.  116. 

326.  Examples  illustrative  of  the  preceding  rules. 

Ex.  1.  It  being  necessary  to  run  a  parallel  to  a  given  in- 
accessible line  AB,  so  as  to  pass  through  a  given  point  C, 
also  inaccessible  and  probably  invisible  from  any  point  in 
the  proposed  line,  I  took  a  base  line  DE  (Fig.  127)  of  18 
chains,  and  at  D  and  E  determined  the  following  angles  of 
position,— viz. :  EDO  =  106°  35';  EDA  =  72°  5';  EDB  = 
21°  20';  DEC  =  26°  50';  DEA  =  61°  20';  and  DEB  =  120° 
45'.  Eequired  the  distance  DG  and  the  angle  DGF ;  also 
the  distance  GC  to  the  given  station. 

Ans.  DG  8.48  ch.,  GC  13.47  ch.,  and  DGF  =  124°  8'  17". 

Ex.  2.  One  side  AB  of  a  tract  of  land  being  inaccessible, 
and  it  being  required  to  run  from  a  given  station  C  a  line 
which  shall  make  an  angle  of  67°  35'  with  that  side,  I 
measured  a'base  line  CD  of  7  chains,  and  took  the  angles 
CDA  =  100°  25';  CDB  =  47°  29';  DCA  =  32°  17';  and 
DCB  ==  90°  3'.  Eequired  the  angle  DCF  which  the  required 
line  makes  with  DC ;  also  the  distance  on  CF  to  the  line 
AB,  and  the  distance  of  the  point  of  intersection  from  A. 
Ans.  DCF  =  49°  10'  20",  CF  =  7.84,  AF  =  2.94. 

Ex.  3.  The  line  AB  not  being  accessible  except  at  its  ex- 
tremities, which  were,  however,  visible  from  each  other,  I 
took  the  angles  as  follow  to  the  points  C  and  D,  whose  dis- 
tance I  had  previously  found  to  be  10.78  chains,  and  found 


SEC.  III.]  OBSTACLES   IN  COMPASS  SURVEYING.  189 

them  to  be  BAD  =  46°  30';  BAG  =  81°  43';  ABC  =  37° 
23';  and  ABD  =  80°  47'.     Eequired  AB. 

Ans.  AB  =  13.76  ch. 

Ex.  4.  To  a  given  inaccessible  line  AB  it  being  required 
to  run  a  perpendicular  which,  shall  pass  through  a  point  P 
also  inaccessible,  I  took  a  base  CD  of  15  chains,  and  mea- 
sured the  angles  as  follow,— viz. :  DCP  =  105°  30';  DCA 
=  256°  50';  DCB  =  326°  42';  PDC  =  38°  50';  PDA  = 
79°  38';  PDB  =  131°  7'.  Required  the  distance  on  DC 
from  D  to  the  proposed  line. 

Ans.  DF  =  14.36. 

Ex.  5.  One  side  AB  of  a  tract  of  land  being  inaccessible, 
and  it  being  required  to  locate  the  adjoining  side  AE,  which 
makes  with  the  former  an  angle  BAE  of  98°  17',  a  base  CD 
of  10  chains  was  measured.  At  C,  the  angle  DCA  was  95° 
and  DCB  =  37°  20'.  At  D,  CDA  was  43°  45',  and  CDB 
=  87°  39'.  Required  the  angle  between  CD  and  a  parallel 
to  AB ;  also  the  distance  on  that  parallel  to  the  point  E  in 
AE,  and  the  distance  AE. 

Ans.  The  parallel  makes  with  CD  the  angle  DCE  =  163° 
57',  CE  =  5.19  ch.,  and  AE  ==  9.89  ch. 

Ex.  6.  In  running  a  random  line  AB  !N~.  87°  E.  towards  a 
point  C,  after  proceeding  7.50  chains  I  came  to  an  impass- 
able swamp.  I  therefore  measured  on  a  perpendicular 
K  3°  W.  4.25  chains,  and  S.  3°  E.  5  chains  to  the  points  D 
and  E  from  which  C  could  be  seen.  At  D,  the  angle  ODE 
was  66°  39',  and  at  E,  DEC  was  67°  25'.  Required  the  dis- 
tance BC,  the  true  course  and  distance  of  AC. 

Ans.  BC  =  10.93  ch.;  AC  =  18.42  ch.;  True  course 
K  88°  26'  E. 


190 


COMPASS   SURVEYING. 


[CHAP.  V. 


SECTION  IV. 

FIELD-NOTES, 

327.  THE  field-notes,  when  the  bearings  are  taken,  are 
recorded  in  various  modes. 

First  Method. — The  simplest  method  is  to  write  them 
after  each  other,  as  ordinary  writing,  thus : — 

Beginning  at  a  limestone  corner  of  James  Brown's  land, 
K  27J°  E.  7.75  chains,  to  a  marked  white-oak.  Thence, 
S.  60J°  E.  10.80  chains,  to  a  limestone,  &c. 

In  recording  the  boundaries,  it  is  well  to  name  the  pro- 
prietors of  the  adjoining  properties.  These  are  always 
inserted  in  deeds  of  conveyance. 

328.  Second  Method. — Bule  three  columns,  as  in  the  ad- 
joining plan :      in   the   first,   insert  the   station ;    in  the 
second,  the  bearing;  and,  in  the  third,  the  distance:  the 
margin  to  the  right  will  serve  for  the  landmarks,  adjoining 
proprietors,  &c.     The  left-hand  page  of  the  book  may  be 
reserved — as  directed  in  Chain  Surveying  —  for  remarks, 
subsidiary  calculations,  &c. 


Sta. 

Bearing. 

Distance. 

Landmarks,  &c. 

1 

2 
3 
4 
5 
6 

JSr.27j°E. 
S.62J°E. 
S.  80°  E. 
S.47J°E. 
S.  54J°W. 
E".37J°W. 

7.75 
10.80 
9.50 
9.37 
8.42 
23.69 

to  a  marked  white-oak. 
"  limestone. 
"        do. 
"  forked  white-oak. 
"  limestone. 
"        do.     the  place  of  beginning. 

329.  Third  Method. — Where  there  are  subsidiary  mea- 
surements,— such  as  offsets,  intermediate  distances,  &c., — 
the  above  method  is  not  convenient,  as  it  requires  a  new 
table  for  each  line  along  which  such  measurements  are 


SEC.  IV.] 


FIELD-NOTES. 


191 


made.  In  such  cases,  the  method  by  columns,  with  mar- 
ginal sketches  of  fences,  streams,  &c.,  is  perhaps  the  best. 
The  notation  for  "False  Stations,"  the  crossing  of  lines, 
streams,  &c.,  (adopted  in  Art.  244,)  may  be  employed  here. 
The  bearing  should  be  inserted  diagonally  in  the  columns, 
and  the  bearings  of  cross  fences,  proof  bearings,  with  the 
offsets,  should  be  recorded  in  the  right  or  left-hand  margin, 
according  as  the  lines  or  points  to  which  they  refer  are  to 
the  right  or  left  of  the  line  being  run. 

Sketches  of  the  adjoining  fences  may  likewise  be  inserted 
in  the  margin,  with  the  distances  to  the  intersections.  By 
this  combination  of  the  columns  and  sketches,  all  the  field- 
work  may  be  recorded  concisely,  luminously,  and  accu- 
rately. 

The  following  notes  of  a  survey  will  illustrate  the 
above : — 


2      I 


Sta.  4 

a       o 

1132 

»        55 

1054 

1        72 

896 

:>        97 

739 

;    75 

48°-$ 

I 

> 

Sta.  3 

Sta.  3 

Limestone  on 

1450 

bank  of  run. 

1030 

^>> 

Sta.  2 

Sta.  2 

a  limestone. 

1344 

(752) 

N.59°10'E. 

Sta.  1 

a  limestone. 

Sta.  1 

1396 

585 

<^o 

Sta.  5 

\ 

Sta.  5 

a  mai 

1740 

corn 

63 

1414 

Phil 

35 

1237 

87 

1016 

rt 

45 

824 

-tfr*x^ 

50 

652 

P*^^ 

551 

0 

462 

75 

295 

75 

Sta.  4 

192 


COMPASS  SURVEYING. 


[CHAP.  V. 


Fig.  141  is  a  plat  of  this  tract. 

Fig.  141. 


SECTION  Y. 


LATITUDES  AND  DEPARTURES, 

DEFINITIONS. 

330.  THE  difference  of  latitude  —  or,  as  it  is  concisely  called, 
the  latitude  of  a  line  —  is  the  distance  one  end  is  farther 
north  or  south  than  the  other. 

It  is  reckoned  north  or  south  according  as  the  bearing  is 
northerly  or  southerly. 

331.  The  difference  of  longitude  or  the  departure  of  a  line  is 
the  distance  one  end  is  farther  east  or  west  than  the  other, 
and  is  reckoned  east  or  west  as  the  bearing  is  easterly  or 
westerly. 

332.  "Where  the  course  is  directly  north  or  south,  the 
latitude  is  equal  to  the  distance,  and  the  departure  is  zero  ; 
but  where  the  bearing  is  east  or  west,  the  latitude  is  zero, 


SEC.  V.]  LATITUDES  AND  DEPARTURES.  193 

and  the  departure  is  equal  to  the  distance.  In  all  other 
cases  the  latitude  and  departure  will  each  be  less  'than  the 
distance,  the  latter  being  the  hypothenuse  of  a  right-angled 
triangle,  of  which  the  others  are  the  legs,  and  the  angle 
adjacent  to  the  latitude  the  bearing.  Thus,  Fi  142 
AB  (Fig.  142)  being  the  line,  AC  is  the  N 
latitude  north,  and  CB  the  departure  east. 

Strictly  speaking,  the  triangle  is  a  right- 
angled  spherical  triangle ;  but  the  deviation 
from  a  plane  is  so  small  as  to  be  abso- 
lutely unappreciable  except  in  lines  of 
great  length.  No  notice  is,  therefore,  taken 
of  the  rotundity  of  the  earth  in  "Land 
Surveying." 

333.  The  latitude,  departure,  and  distance  being  the  sides  of 
a  right-angled  triangle,  of  which  the  bearing  is  one  of  the  acute 
angles,  any  two  of  these  may  be  found  if  the  others  are  known. 

1.  Given  the  bearing  and  distance,  to  find  latitude  and 
departure. 

As  radius  :  cosine  of  bearing  : :  distance  :  latitude ; 
and  as  radius  :    sine   of  bearing  : :  distance  :  departure. 

2.  Given  the  latitude  and  departure,  to  find  the  bearing 
and  distance. 

As  latitude  :  departure  : :  radius  :  tangent  of  bearing. 
As  cosine  of  bearing  :  radius  : :  latitude  :  distance. 

3.  Given  the  bearing  and  departure,  to  find  the  distance 
and  latitude. 

As  sine  of  bearing  :  radius  : :  departure  :  distance. 

As  radius  :  cotangent  of  bearing  : :  departure  :  latitude. 

4.  Given  the  bearing  and  latitude,  to  find  the  distance 
and  departure. 

As  cosine  of  bearing  :  radius  : :  latitude  :  distance. 
As  radius  :  tangent  of  bearing  : :  latitude  :  departure. 

13 


194  COMPASS   SURVEYING.  [CHAP.  V 

5.  Given  the  distance  and  latitude,  to  find  the  bearing 
and  departure. 

As  distance  :  latitude  : :  radius  :  cosine  of  bearing. 
As  radius  :  sine  of  bearing  : :  distance  :  departure. 

6.  Given  the  distance  and  departure,  to  find  the  bearing 
and  latitude. 

As  distance  :  departure  : :  radius  :  sine  of  bearing. 
As  radius  :   cosine  of  bearing  : :  distance  :  latitude. 

EXAMPLES. 

Ex.  1.  Giving  the  bearing  and  distance  of  a  line  1ST.  56£° 
TV.  37.56  chains,  to  find  the  latitude  and  departure. 

Ans.  Lat.  20.87  K;  Dep.  31.23  W. 

Ex.  2.  Given  the  difference  of  latitude  36.17  K,  and  the 
distance  52.95,  to  find  the  bearing  and  departure,  east. 

Ans.  Bearing  =  K  46°  55'  E.;  Dep.  =  38.67. 

Ex.  3.  Given  the  difference  of  latitude  19.25  K,  and  the 
departure  26.45  W,,  to  find  the  bearing  and  distance. 

Ans.  Bearing  =  K  53°  57'  W. ;  dist.  =  32.71. 

Ex.  4.   Given  the  bearing  S.  33|°  W.,  and  the  departure 
18.33  chains,  to  find  the  distance  and  difference  of  latitude. 
Ans.   Dist.  =  33.21  ch.;   Lat.  =  27.69  S. 

334.  Traverse  Table.  The  traverse  table  contains  the 
latitudes  and  departures  for  every  quarter  degree  of  the 
quadrant  to  all  distances  up  to  ten.  From  these,  the  lati- 
tude and  departure,  corresponding  to  any  bearing  and  dis- 
tance, may  readily  be  found  by  the  following  rule : — 

If  the  distance  be  not  greater  than  ten. — Seek  the  degrees  at 
the  top  or  bottom  of  the  table  according  as  their  number  is 
less  or  greater  than  45°,  and  in  the  columns  marked  Lati- 
tude and  Departure,  opposite  to  the  distance,  will  be  found 
the  latitude  and  departure.  If  the  degrees  are  found  at  the 
bottom  of  the  table,  the  name  of  the  column  is  there  like- 
wise. For  all  degrees  less  than  forty  five,  the  left-hand 


SEC.  V.]  LATITUDES  AND  DEPARTURES.  195 

column  is  the  latitude,  but  the  departure,  for  those  greater 
than  45°. 

If  the  distance  be  more  than  ten,  and  consist  of  whole  tens. — 
Take  out  the  number  from  the  table  as  before,  and  remove 
the  decimal  point  as  many  places  to  the  right  as  there  are 
ciphers  at  the  right  of  the  distance  in  the  table. 

If  the  distance  is  not  composed  simply  of  tens. — Take  from 
the  table  the  latitude  and  departure  corresponding  to  every 
figure,  removing  the  decimal  point  as  many  places  to  the 
right  or  to  the  left  as  the  digit  is  removed  to  the  left  or  the 
right  of  the  unit's  place,  and  take  the  sum  of  the  results. 

EXAMPLES. 

Ex.  1.  Required  the  latitude  and  departure  of  a  line 
bearing  1ST.  37J°  E.  8  chains. 

Opposite  to  8  chains,  under  the  degrees  37J,  are  found, — 

Lat.  6.3680,  Dep.  4.8424. 
The  latitude  and  departure  required  are,  therefore, 

6.37  K,  4.84  E. 

If  the  distance  had  been  80  chains,  the  latitude  and  de- 
parture would  have  been 

63.68  K,  48.42  E. 

Ex.  2.  Required  the  latitude  and  departure  of  a  line  run- 
ning S.  63J°  E.  75  chains. 

70  ch.  Lat.  31.234  Dep.  62.645 

5  «  2.231  4.475 

33.465  67.120 

Hence  the  result  is        Lat.  33.46  S. ;       Dep.  67.12  E. 

Ex.  3.  Required  the  latitude  and  departure  of  a  line  run- 
ning K  35f  °  W.  58.65  chains. 

50  ch.  Lat.  40.579  Dep.  29.212 

8  "  6.493  4.674 

.6  487  351 

.05  41^  29 

Lat.  47.600  K         Dep.  34.266  W. 


196  COMPASS   SURVEYING.  [CHAP.  V. 

Ex.  4.  What  are  the  latitude  and  departure  of  a  line  bear- 
ing S.  63J°  W.  27.49  chains? 

Ans.  Lat.  12.27  S.;  Dep.  24.60  W. 

Ex.  5.  What  are  the  latitude  and  departure  of  a  line  IN".  55f  ° 
E.  27  chains  ?  Ans.  Lat.  15.20  ET.;  Dep.  22.32  E. 

Ex.  6.  "What  are  the  latitude  and  departure  of  a  line  bear- 
ing K  84f°  E.  123.56  chains? 

Ans.   Lat.  11.31  ST. ;  Dep.  123.04  E. 

Ex.  7.  What  are  the  latitude  and  departure,  the  bearing 
and  distance  being  S.  24f  °  W.  97.56  chains  ? 

Ans.   Lat.  88.60  S. ;  Dep.  40.84  W. 

335.  When  the  bearing  is  given  to  minutes.  Take  out  the 
numbers  in  the  table  for  the  quarter  degrees  between  which 
the  minutes  fall.  Then  say, — 

As  15  minutes  is  to  the  excess  of  the  given  number  of 
minutes  above  the  less  of  the  two  quarters,  so  is  the  dif- 
ference of  the  numbers  in  the  table  to  a  fourth  term,  which 
must  be  subtracted  from  the  number  corresponding  to  the 
less  of  the  two  quarters  if  the  quantity  is  a  latitude,  but 
added  if  it  is  a  departure. 

Thus,  supposing  the  line  were  N.  41°  18'  E.  43.27  chains. 
Take  the  difference  between  the  latitude  for  41  J°  and  that 
for  41  J-%  and  say, — 

As  15'  is  to  the  difference  between  41  J°  and  41°  18',  or  3', 
so  is  the  difference  between  the  latitudes  to  the  correction 
for  3'.  This  correction  subtracted  from  the  latitude  for 
41  J°  will  give  the  latitude  required. 

Do  the  same  with  the  departure,  except  that  the  correc- 
tion found  as  above  must  be  added  to  the  departure  for  41  J°. 

In  the  example,  we  have  for  the  distance  40  in  the 
column  for 

41J°  the  Lat.  30.074  Dep.  26.374 

41J°  29.958  26.505 

Differences  .116  .131 

Then,       As  15'  :  3'  : :  .116  :  .023,  correction  of  latitude ; 
and,          As  15'  :  3'  : :  .131  :  .026,  correction  of  departure. 


SEC.  V.]  LATITUDES  AND  DEPARTURES.  197 

The  corrected  latitude  and  departure  for  41°  18',  distance 
40  chains,  are  Lat.  30.051.,  Dep.  26.400. 

In  like  manner,  the  latitudes  and  departures  for  each  of 
the  remaining  figures  may  be  calculated,  being  as  below : — 

For             40  ch.  Lat.  30.051  Dep.  26.400 

3  "  2.254  1.980 

.2  150  132 

.07  53  46 


32.508  K  28.558  E. 

There  will  rarely  be  any  calculation  necessary  for  the 
decimal  figures  of  the  distance,  as  the  variation  caused  by 
a  quarter  of  a  degree  will  seldom  change  more  than  a  unit 
any  of  the  figures  that  need  be  retained. 

Ex.  1.  The  bearing  and  distance  being  1ST.  76°  42'  E.  39.76 
chains,  to  find  the  difference  of  latitude  and  departure. 

Ans.  Lat.  9.147  K ;  Dep.  38.694  E. 

Ex.  2.  Given  the  bearing  and  distance  S.  37°  9'  E.  63.45 
chains,  to  find  the  difference  of  latitude  and  departure. 
Ans.   Lat.  50.573  S. ;  Dep.  38.317  E. 

Ex.  3.  Required  the  difference  of  latitude  and  departure 
of  a  line  running  S.  29°  17'  E.  123.75  chains. 

Ans.  Lat,  107.937  S. ;  Dep.  60.529  E. 

336.  By  Table  of  Natural  Sines  and  Cosines.  The  differ- 
ence of  latitude  and  departure,  when  the  bearing  is  given 
to  minutes,  is  more  readily  found  from  the  table  of  natural 
sines  and  cosines  than  from  the  traverse  table.  The  dif- 
ference of  latitude  and  departure  are  the  cosine  and  the 
sine  of  the  bearing  to  a  radius  equal  to  the  distance. 
Therefore,  to  find  the  difference  of  latitude  and  departure 
of  a  line,  take  out  the  natural  cosine  and  sine  of  the  bear- 
ing, and  multiply  them  by  the  distance. 

Ex.  1.  Required  the  difference  of  latitude  and  departure 
of  a  line  bearing  N.  41°  18'  E.  43.27  chains. 


COMPASS  SURVEYING.  [CHAP.V. 

41°  18'  Cosine  .75126               Sine  66000 

Dist.  Diff.  Lat.  Dep. 

40  ch.  30.0504  26.4000 

3    «  2.2538  1.9800 

.2  1503  1320 

.07  526  462 


Lat.  32.5071  N.        Dep.  28.5582  E. 

The  result  by  this  method  may  be  depended  on  to  the 
third  decimal  figure,  unless  the  distance  is  several  hundred 
chains,  and  then  it  will  rarely  affect  the  second  decimal 
figure. 

Ex.  2.  Eequired  the  latitude  and  departure  of  a  line 
K  29°  38'  E.  26.47  chains. 

29°  38'  Cosine  .86921  Sine.49445 


20  ch.  17.3842  9.8890 

6  «  5.2153  2.9667 

.4  .3477  1978 

.07  608  346 


Lat,  23.0080  K       Dep.  13.0881  E. 

The  calculation  need  not,  in  general,  be  carried  beyond 
the  third  decimal  place.  In  the  above  example  the  work 
would  then  stand  thus : 

29°  38'  Cosine  .86921  Sine.49445 


20  ch.  17.384  9.889 

6  "  5.215  2.967 

.4  348  198 

.07  61  34 

Lat.  23.008  ST.  Dep.  13.088  E. 

Ex.  3.  Eequired  the  latitude  and  departure  of  a  line  bear- 
ing S.  56°  V  E.  63.48  chains. 

Ans.    Lat,  35.39  S. ;  Dep.  52.70  E. 

Ex.  4.  Eequired  the  latitude  and  departure  of  a  line  bear- 
ing K  52°  49'  W.  136.75  chains. 

Ans.  Lat.  82.65  K;  Dep.  108.95  W. 


SEC.  V.]  LATITUDES  AND  DEPARTURES.  199 

Ex.  5.  Given  the  bearing  and  distance  S.  23°  47'  W. 
13.62  chains,  to  find  the  latitude  and  departure. 

Ans.    Lat.  12.46  S.;  Dep.  5.49  W. 

337.  Test  of  the  Accuracy  of  the  Survey.  When  the 
surveyor  has  gone  round  a  tract,  and  has  come  back  to  the 
point  from  which  he  started,  it  is  self-evident  that  he  has 
travelled  as  far  in  a  southerly  direction  as  he  has  in  a 
northerly,  and  as  far  easterly  as  westerly. 

His  whole  northing  must  equal  his  whole  southing,  and 
his  whole  easting  equal  his  whole  westing.  If  then  the 
north  latitudes  are  placed  in  one  column  and  the  south  lati- 
tudes in  another,  the  sum  of  the  numbers  in  these  columns 
will  be  equal,  provided  the  bearings  and  distances  are 
correct.  So  also  the  columns  of  departures  will  balance 
each  other. 

Owing  to  the  unavoidable  errors  in  taking  the  measure- 
ments, and  also  to  the  fact  that  the  bearings  are  generally 
taken  to  quarter  degrees,  this  exact  balancing  rarely  occurs 
in  practice.  When  the  sums  are  nearly  equal,  we  may 
attribute  the  error  to  the  want  of  precision  in  the  instru- 
ments ;  but,  if  the  error  is  considerable,  a  new  survey  should 
be  made.  * 

It  not  unfrequently  happens  that  the  mistake  has  been 
made  on  a  single  side.  This  can  often  be  detected  by 
taking  the  errors  of  latitude  and  departure,  and  calculating 
or  estimating  the  bearing  of  a  line  which  should  produce 
such  an  error  by  a  mismeasurement  of  its  length  or  a  mis- 
take in  its  bearing.  A  little  ingenuity  will  then  frequently 
enable  the  surveyor  to' judge  of  the  probable  position  of  the 
error,  and  thus  obviate  the  necessity  of  a  complete  resurvey 
of  the  tract. 

It  is  laid  down  as  a  rule  by  some  good  surveyors  tha,t  an 
error  of  one  link  for  every  five  chains  in  the  whole  distance 
is  the  most  that  is  allowable.  When  the  transit  or  theodo- 
lite is  used,  a  much  closer  limit  should  be  drawn.  One 
link  for  ten  or  fifteen  chains  is  quite  enough,  unless  the 
ground  is  very  difficult.  Every  surveyor  will,  however, 


200  COMPASS   SURVEYING.  [CHAP.  V. 

form  a  rule  for  himself,  dependent  on  his  experience  of  the 
precision  to  which  he  usually  obtains.  A  young  surveyor 
should  set  a  high  standard  of  excellence,  as  he  will  find  this 
to  be  a  very  good  method  of  making  himself  accurate.  If 
he  begins  by  being  satisfied  with  poor  results,  the  chances 
are  that  he  will  never  attain  to  a  high  rank  in  his  profession. 

338.  Correction  of  Latitudes  and  Departures. 

When  the  northings  and  southings,  or  the  eastings  and 
westings,  do  not  balance,  the  error  should  be  distributed 
among  the  sides  before  making  any  calculations  dependent 
upon  them. 

The  usual  mode  of  distributing  the  error  is  to  apply  to 
each  line  a  portion  proportioned  to  its  length. 

Rule  a  table,  and  head  the  columns  as  in  the  adjoining 
example.  Take  the  latitudes  and  departures  of  the  several 
sides,  and  place  them  in  their  proper  columns. 

Take  the  difference  between  the  sum  of  the  northings 
and  that  of  the  southings.  The  result  is  the  error  in  lati- 
tude, and  should  be  marked  with  the  name  of  the  less  sum. 

Do  the  same  with  the  eastings  and  westings :  the  result  is 
the  error  in  departure,  of  the  same  name  as  the  less  sum. 

Divide  the  error  of  latitude  by  the  sum  of  the  distances : 
the  quotient  is  the  correction  for  1  chain. 

Multiply  the  correction  for  1  chain  by  the  number  of 
chains  in  the  several  sides :  the  products  will  be  the  correc- 
tions for  those  sides,  which  may  be  set  down  in  a  column 
prepared  for  the  purpose,  or  at  once  applied  to  the 
latitude. 

Operate  the  same  way  with  the  error  in  departure,  to 
obtain  the  corrections  of  departure  of  the  several  sides. 

The  corrections  are  of  the  same  name  as  the  errors. 

The  corrections  above  found  are  to  be  applied  by  adding 
them  when  of  the  same  name,  but  subtracting  if  of  different 
names. 

If  one  side  of  a  tract  is  hilly,  or  otherwise  difficult  to 
measure,  a  larger  share  of  the  error  should  be  attributed  to 
that  side. 

When  a  change  of  bearing  of  a  long  side  will  lessen  the 


SEC.  V.] 


LATITUDES  AND   DEPARTURES. 


201 


error,  this  change  should  be  made,  especially  if  the  survey 
was  made  with  a  compass. 

The  corrections  may  be  made  in  the  original  columns  by 
using  red  ink.     New  columns  are,  however,  to  be  preferred. 

Ex.  1.  Given  the  bearing  and  distances  as  follows,  to  find 
the  corrected  latitudes  and  departures. 


1 

K43J°W. 

28.43 

2 

ff.  29}°  E. 

30.55 

3 

S.  80°  E. 

28.74 

4 

East. 

40.00 

5 

S.  10J°  E. 

23.70 

6 

S.  64°  W. 

25.18 

7 

K63f°W. 

20.82 

8 

S.  57J°W. 

31.65 

Bearings. 

Dirt. 

N. 

S. 

E. 

W. 

Cor. 

N. 

Cor. 
W. 

N. 

S. 

E. 

W. 

1   N.43^°W. 

28.43 

20.62 

19.57 

.01 

20.62 

19.58 

2    N.29%°E. 

30.55 

26.52 

15.16 

.02 

26.52 

15.14 

3     S.  80°  E. 

28.74 
40.00 

4.99 

28.30 

TbT 

.02 
^02~ 

4.99 

28.28 

4        East. 

40.00 

.01 

39.98 

5    S.10^°  E. 

23.70 

23.32 

4.22 

.01 

23.32 

4.21 

6     S.  64°  W. 

25.18 

11.04 

22.63 

.01 

11.04 

22.64 

7    N.63%°W. 

20.82 

9.21 

18.67 

.01 

9.21 

18.68 

8    S.57%°W. 

31.65 

17.01 

26.69 

.02 

17.01 

26.71 

229.07      56.35     56.36     87.68      87.56     .01    .12     56.36     56.36     87.61     87.61 
56.33     87.56 
Er.N.   .01         .12Er.W. 

Ex.  2.  Correct  the  latitudes  and  departures  from  the  fol- 
lowing notes:— 1.  S.  49°  W.  12.93  ch.;  2.  S.  88°  W.  13.68 
ch. ;  3.  K  25J°  W.  14.09  ch. ;  4.  K  43J°  E.  14.70  ch. ;  5. 
K  12J°  W.  17.95  ch. ;  6.  ST.  88J °  E.  17.68  ch. ;  7.  S.  36J°  E. 
35.80 ch.;  8.  S.  77J°  W.  16.15  ch. 

Ans.  1.  S. 8.48,  W.  9.76;  2.  S.. 48,  W.  13.67;  3.  K  12.73, 
W.  6.01;  4.  K  10.70,  E.  10.07;  5.  K  17.51,  W.  3.88 ;  6.  K 
38,  E.  17.69;  7.  S.  28.79,  E.  21.30;  8.  S.  3.57,  W.  15.74. 


?,03 


COMPASS   SURVEYING. 


[CHAP.  V. 


SECTION  VI. 
PLATTING  THE  SURVEY,* 

339.  With  the  Protractor.  First  Method. — DRAW  a  line 
!N"S,  on  any  convenient  part  of  the  paper,  to  represent  the 
meridian. 

Place  the  protractor  with  its  straight  edge  to  this  line, 
and  its  arc  turned  to  the  right  if  the  bearing  be  easterly, 
but  to  the  left  if  it  be  westerly,  and  with  a  fine  point  mark 
off  the  number  of  degrees.  Draw  a  straight  line  from  the 
centre  to  this  point,  and  on  it  lay  off  Fis- 

the  distance.  The  point  2  (Fig.  143) 
will  thus  be  determined.  Through  2 
draw  a  line  parallel  to  N"  S.  Place  the 
protractor  with  its  centre  at  2  and  its 
straight  side  coincident  with  the  me- 
ridian, and  prick  off  the  degrees  in 
the  bearing  of  the  second  side.  Join 
this  point  to  2,  and  on  the  line  thus 
determined  lay  off  2.3  equal  to  the 
second  side.  Through  3  draw  another 
meridian ;  and  so  proceed  until  all  the 
bearings  and  distances  have  been  laid  down. 

"When  the  last  line  has  been  platted,  it  should  end  at  the 
starting  point:  if  it  does  not,  either  the  notes  are  incorrect 
or  an  error  has  been  made  in  the  platting 

The  proper  position  of  the  protractor  after  the  first  may 
be  determined  without  drawing  meridians,  by  placing  the 
centre  at  the  point  and  turning  the  protractor  until  the 
number  of  degrees  in  the  bearing  of  the  last  line  coin- 
cides with  that  line.  Its  position  is  then  parallel  to  the 
former  one,  and  the  bearing  of  the  next  line  may  be 
pricked  off. 

This  method  is  the  one  commonly  employed.  It  has, 
however,  the  disadvantage  of  accumulating  errors,  since  any 
mistake  in  laying  down  the  bearing  of  one  line  will  alter 


*  Various  hints  in  this  section  have  been  derived  from  Gillespie's 
Surveying." 


Land 


SEC.  VI.] 


PLATTING  THE   SURVEY. 


203 


both  the  direction  and  position  of  -every  subsequent  line  on 
the  plat. 

The  figure  is  the  plat  from  the  following  field-notes : — 
1.  K  27  J°  E.  7.T5;  2.  S.  60J°  E.  10.80;  3.  S.  8°  E.  9.50; 
4.  S.  47J°  E.  9.37;  5.  S.  54J°  W.  8.42;  6.  K  37J°  "W.  23.69. 

340.  Second  Method. — Draw  a  number  of  parallel  lines  to 
represent  meridians.  They  may  be  equidistant  or  not. 
The. faint  lines  on  ruled  paper  will  answer  very  well. 

Select  any  convenient  point  for  Fig.  144 

a  place  of  beginning,  and  draw  the 
line  AB  (Fig.  144)  for  the  first  side. 
Place  the  protractor  so  that  its 
centre  shall  be  on  one  of  the  me- 
ridians, and  turn  it  until  the  num- 
ber of  degrees  in  the  next  side 
coincides  with  the  same  meridian, 
as  at  C :  slip  it  down  the  line, 
maintaining  the  coincidence  of  the 
centre  and  degree  mark  with  the  meridian,  until  the 
straight  side  passes  through  the  point  Draw  a  line  along 
this  side.  It  will  be  the  direction  of  the  required  line,  on 
which  lay  off"  the  given  distance.  So  continue  until  all  the 
sides  have  been  platted.  The  figure  will  close,  if  the  work 
is  properly  done. 

This  method  is  quite  as  accurate  as  the  last,  and  admits 
of  very  rapid  execution. 


/ 
\ 


\ 


\ 


341.  By  a  Scale  of  Chords.  With 
a  radius  equal  to  the  chord  of  60° 
describe  a  circle  near  the  middle  of 
the  paper.  Through  its  centre  0  (Fig. 
145)  draw  a  line  RTS  to  represent  the 
meridian.  Lay  off  from  the  north  and 
south  points  the  different  bearings,  5 
marking  them  1,  2,  &c.  Through 
A,  any  convenient  point,  draw  AB 
parallel  to  0.1,  and  on  it  lay  off  AB 
equal  to  the  length  of  the  first  side 


Fig.  145. 


204 


COMPASS   SURVEYING. 


[CHAP.  Y. 


taken  from  any  convenient  scale.  Through  B  draw  BC 
parallel  to  0.2:  on  it  lay  off  BC  equal,  to  the  second  side. 
Through  C  draw  CD  parallel  to  0.3;  and  so  proceed  till  all 
the  lines  have  been  platted. 

With  an  accurate  scale  of  chords  of  a  good  size,  this 
method  is  probably  preferable  to  either  of  the  others.  The 
scale  on  the  rule  sold  with  cases  of  instruments,  however, 
is  so  small  that  no  great  precision  can  be  obtained  by  its 
use.  It  is  still,  however,  preferable  to  the  other  methods  if 
the  protractor  in  similar  cases  of  instruments  is  employed. 

342.  By  a  Table  of  Natural  Sines.  The  sine  of  any 
arc  is  equal  to  half  the  chord  of  twice  that  arc,  or  to  the 
chord  of  twice  the  number  of  degrees  on  a  circle  of  half 
the  radius.  "We  may  therefore  use  a  table  of  natural  sines 
to  lay  off  angles.  Its  use  in  protracting  a  survey  is  ex- 
plained below. 

(Fig.    146) 

the   paper 

to  5  on  a 

This  scale 


Fig.  146. 

N 


Describe  a  circle 
about  the  centre  of 
with  a  radius  equal 
scale  of  equal  parts, 
should  be  taken  as  large  as  con- 
venient.  Through  its  centre  A 
draw  JSTS  to  represent  the  me- 
ridian, and  cross  the  circle  at  the 
points  marked  60°,  with  the 
centres  N"  and  S,  and  radius  equal  to  that  of  the  circle:  also 
draw  EW  perpendicular  to  !N"S.  The  points  marked  30° 
may  be  obtained  by  crossing  the  circle  with  the  compasses 
opened  to  the  radius  and  one  leg  at  E  and  W. 

A  skeleton  protractor  is  thus  formed,  having  the  E"orth, 
South,  East,  and  West  points,  as  well  as  the  30°  and  60° 
points,  accurately  laid  down. 

Commencing  with  the  first  bearing,  which  in  the  figure  is 
N".  27J  E.,  divide  it  by  2,  and  from  the  table  of  natural 
sines  take  out  the  sine  of  the  quotient  13°  45'.  It  is  found 
to  be  2.3769,  the  decimal  point  being  removed  1  place  to 
the  right.  Take  this  distance  2.38  from  the  scale  of  equal 
parts,  and  lay  it  off  from  N"  to  1. 


SEC.  VI.]  PLATTING  THE  SURVEY.  205 

The  second  bearing  is  S.  60J°  E.  The  half  of  J°  is  15' : 
the  sine  of  this  is  0.0436.  Lay  off  .04  from  60°  to  2. 

The  third  bearing  is  S.  8°  E. :  the  sine  of  4°  is  0.6976.  Lay 
off  .70  from  S.  towards  E. :  the  point  3  is  thus  determined. 

The  fourth  is  S.  47J°  E.,  which  exceeds  30°  by  17J° :  the 
half  of  17J°  is  8°  45',  of  which  the  sine  is  1.5212.  1.52 
laid  off  from  30  towards  E.  determines  the  point  4. 

An  accurate  protractor  is  thus  formed  on  the  paper,  con- 
taining all  the  bearings  in  the  field-notes.  The  subsequent 
work  will  be  as  in  last  article. 

343.  By  a  Table  of  Chords.     Instead  of  a  table  of 
natural  sines,  a  table  of  chords,  when  it  can  be  procured,  is 
more  convenient. 

Prepare  a  circle,  as  in  last  article,  with  the  E".,  S.,  E.,  W., 
and  the  30°  and  60°  points,  the  radius  being  10,  taken 
from  a  scale  of  equal  parts. 

Take  from  the  table  the  chord  of  the  number  of  degrees, 
or  of  its  excess  above  30°  or  60°,  and  lay  it  off  from  the 
proper  point,  as  directed  in  last  article:  an  accurate  pro- 
tractor is  thus  formed  on  the  paper,  and  the  work  proceeds 
as  before. 

The  object  in  determining  the  30°  and  60°  points  is  to 
avoid  the  necessity  of  laying  off  long  distances.  "When  the 
compasses  are  much  stretched,  the  points  strike  the  paper 
very  obliquely,  and  are  apt  to  sink  in  so  as  to  make  the  dis- 
tance laid  off  slightly  too  short. 

This  method  is  preferable  to  any  of  those  which  precede 
it :  it  is  only  to  be  excelled  by  the  one  next  given. 

344.  By  Latitudes  and  Departures. 

"Where  the  latitudes  have  been  calculated  and  balanced, 
they  afford  the  most  convenient  and  accurate  means  of 
platting  the  survey. 

Rule  five  columns,  heading  them  Sta.,  N".,  S.,  E.,  "W", 
Commencing  at  any  convenient  station,  place  the  latitude 
and  departure  of  the  side  beginning  at  this  station  oppo- 
site the  next  station  in  the  table,  and  in  their  appropriate 
columns.  When  the  latitude  set  down  is  of  the  same  name 


206 


COMPASS  SURVEYING. 


[CHAP.  V. 


as  that  of  the  next  side,  add  them  together,  and  place  the 
result  in  the  proper  column  of  latitudes  opposite  the  next 
side.  But  if  they  be  of  different  names,  take  their  differ- 
ence, and  place  it  in  the  column  of  the  same  name  as  the 
greater.  Proceed  in  the  same  way  with  this  result  and  the 
next  latitude,  and  so  continue  till  all  the  latitudes  have 
been  used.  The  results  will  be  the  latitude  of  the  stations 
opposite  which  they  are  placed,  all  counted  from  the  point 
at  which  we  commenced. 

Proceed  in  the  same  manner  with  the  departures.  Thus, 
if  it  were  required  to  plat  the  survey  of  which  the  field- 
notes  are  given  Ex.  1,  Art.  338,  we  have  the  latitudes  and 
departures,  as  in  the  following  table.  (See  the  example  re- 
ferred to): — 


Sta. 

N. 

s. 

E. 

w. 

1 

20.62 

19.58 

2 

26.52 

15.14 

3 

4.99 

28.28 

4 

.01 

39.98 

5 

23.32 

4.21 

6 

11.04 

22.64 

7 

9.21 

18.68 

8 

17.01 

26.71 

Preparing  a  table  as  above  directed,  and  beginning  at  the 
fourth  station,  the  total  latitudes  .and  departures  will  be  as 
below : — 


Sta. 

N. 

s. 

E. 

w. 

1 

42.15 

23.84 

2 

21.53 

43.42 

3 

4.99 

28.28 

4 

00 

0.00 

5 

.01 

39.98 

6 

23.31 

44.19 

7 

34.35 

21.55 

8 

25.14 

2.87 

;.  VI.] 


PLATTING  THE  SURVEY. 


207 


The  latitude  of  the  fourth  side  is  .01  K  This  is  put  in 
the  column  headed  north,  opposite  the  fifth  station.  The 
next  latitude  being  south,  take  the  difference  23.31 ;  place 
it  in  the  south:  add  23.31  and  11.04,  both  being  south,  and 
we  have  34.35  S.  Subtract  from  this  9.21  K  leaves  25.14  S. 
This,  added  to  1T.01  S.,  gives  42.15  S.  Subtract  20.62  K 
leaves  21.53  S.;  21.53  S.  from  26.52  K,  the  next  latitude, 
leaves  4.99  K  Finally,  4.99  K  and  4.99  S.  cancel,  leaving 
0  for  the  latitude  of  the  fourth  station.  In  the  same  man- 
ner we  find  the  total  departures. 

As  the  latitude  and  departure  of  the  station  with  which 
we  begin  are  zero,  the  work  proves  itself.  It  is  usual  to 
begin  with  the  first  side. 

The  table  having  been  prepared  as  above,  draw  on  any 
convenient  part  of  the  paper  a  meridian  line,  !N"S,  (Fig.  147,) 
and  take  any  point  E  for  the  starting  point.  From  this 


Fig.  147. 

N 


point,  lay  off  the  several  total  latitudes  contained  in  the  table 
above  or 'below  the  point  as  the  latitude  is  north  or  south,  and 
number  them  according  to  the  station  to  which  they  are  op- 
posite in  the  table. 

Through  these  points  draw  perpendiculars  to  the  me- 
ridian, and  make  them  equal  to  the  several  total  de- 
partures,— laying  the  distance  to  the  right  hand  if  the 
departure  be  east,  but  to  the  left  if  it  be  west.  The  cor- 


208  COMPASS  SURVEYING.  [CHAP.  V. 

ners  will  thus  be  determined.    "When  these  are  joined,  the 
plat  will  be  completed. 


SECTION  VII. 

PROBLEMS  IN  COMPASS  SURVEYING, 

345.  Problem  1. — GIVEN  the  bearing  of  one  side,  and 
the  deflection  of  the  next,  to  determine  its  bearing. 

If  the  given  bearing  is  northeasterly  or  southwesterly,  add 
the  deflection  if  it  is  to  the  right  hand.  If  the  sum  exceeds 
90°,  take  its  supplement,  and  change  north  to  south,  or 
south  to  north. 

If  the  deflection  is  to  the  left  hand,  subtract  it  from  the 
bearing ;  but  if  it  is  greater  than  the  bearing  from  which  it 
is  to  be  subtracted,  take  the  difference,  and  change  east  to 
west,  or  west  to  east. 

"When  the  given  bearing  is  northwesterly  or  southeasterly, 
add  the  left-hand  and  subtract  the  right-hand  defections,  ap- 
plying the  same  rules  as  above. 

EXAMPLES. 

Ex.  1.  Given  AB  (Fig.  148)  K".  37°  E.,  Kg.i«. 

and  the  deflection  of  the  next  side  43°  c'  *        D 

15' to  the  right. 

BD  =  K  37°        E.    w _ 

DBC=      43°  15' 

"Whence  BC  is  K  80°  15'  E.  A      g 

Ex.  2.  Given  AB  K  37°  E.,  and  the  deflection  of  BC' 
43°  15'  to  the  left. 

BD  =  K  37°        E. 
DBC'  =      43°  15' 
Whence  BC'  is  N.    6°  15'  W. 


SEC.  VII.]  PROBLEMS  IN  COMPASS  SURVEYING.  209 

Ex.  3.  Given  the  bearing  of  AB,  K  39°  W.,  and  BC  de- 
flects to  the  left  75°  26':  required  the  bearing  of  BC. 

Ans.   S.  65°  34'  W. 

Ex.  4.  Given  the  bearing  of  a  line  S.  63°  29'  E.,  and 
the  deflection  of  the  next  29°  17'  to  the  right :  required  its 
bearing. 

Ans.   S.  34°  12'  E. 

Ex.  5.  The  bearing  of  one  line  being  S.  34°  12'  E.,  and 
the  deflection  of  the  next  75°  32'  to  the  right:  required  its 
bearing. 

Ans.   S.  41°20'W. 

346.  Problem  2.  —  To  determine  the  angle  of  deflection 
between  two  courses. 

1.  If  the  lines  run  between  the  same  points  of  the  com- 
pass, take  the  difference  of  their  bearings. 

2.  If  they  run  between  points  directly  opposite,  subtract 
the  difference  of  the  bearings  from  180°. 

3.  If  they  run  from  the  same  point  towards  different 
points,  add  the  bearings. 

4.  If  they  run  from  different  points  towards  the  same 
point,  take  the  sum  of  the  bearings  from  180°. 

EXAMPLES. 

Ex.  1.  AB  (Fig.  149)  runs  S.  56°  W., 
and  BC  S.  25°  W. :  required  the  de- 
flection. 

w 
56° 

25° 
Deflection  31°  to  the  left. 

14 


210 


COMPASS  SURVEYING. 


[CHAP.  V. 


Ex.  2.  Given  AB  (Fig.  150)  K  46  W., 
and  BC   S.  79°  E. :    required  the  de-      D 
flection. 

1ST.  46°  W.  w 

S.  79°  E. 
33° 

180° 


Eig.150. 


AB 
BC 
ABC 

DBC 


147°  =  deflection  to  the  right. 


Ex.  3.  Given  AB  (Fig.  151)  K  39°  E., 
and    BC  K   63°  W.,  to  find  the  de-      c 
flection. 

AB  N.  39°  E. 

BC  K  63°  W. 


Fig.  151. 


DBC 


102°  =  deflection  to  the  left. 


Ex.4.  Given  AB  (Fig.  152)8.  82°  E., 
and  BC  BT.  67°  E.,  to  find  the  de- 
flection. 

AB  S.   82°  E. 

BC  K  67°  E. 


Fig.  152. 


DBC 


149° 

180° 
31°  =  deflection  to  the  left. 


Ex.  5.  The  bearing  of  a  line  is  K  46°  30'  E.,  and  that  of 
,the  next  S.  63°  29'  "W. :  required  the  deflection. 

Ans.   163°  1'  to  the  left. 

Ex.  6.  What  is  the  deflection  in  passing  from  a  course 
S.  63°  W.  to  one  K  29°  W.? 

Ans.    88°  to  the  right. 

Ex.  7.  "What  is  the  deflection  in  passing  from  a  course 
N.  82J°W.to  one  K  29J°  W.? 

Ans.    53J°  to  the  right. 


347,  Angle  between  lines.    If  the  angle  between  two 


SEC.  VII.]  PROBLEMS  IN  COMPASS  SURVEYING.  211 

lines  is  required,  reverse  the  first  bearing,  and  apply  the 
above  rules. 

EXAMPLES. 

Ex.  1.  Given  AB  K  87°  E.,  and  BC  S.  25°  W.,  to  find 
the  angle  ABC.  Ans.  ABC  =  62°. 

Ex.  2.  Given  AB~S.  63°  E.,  and  BC  K  56°  E. :  required 
the  angle  ABC.  Ans.  ABC  =  119°. 

Ex.  3.  Given  CD  K 15°  W., and  DE  1ST.  56°  W.:  required 
the  angle  CDE.  Ans.  CDE  =  139°. 

Problem  3. — To  change  the  bearings  of  the  sides  of  a 
survey. 

348.  It  is  frequently  useful  to  change  the  bearings  of  a 
survey  so  as  to  determine  what  they  would  be  if  one  side 
were  made  a  meridian.  This  change  is  made  on  the  sup- 
position that  the  whole  plat  is  turned  around  without  alter- 
ing the  relative  positions  of  the  sides.  Every  bearing  will 
thus  be  altered  by  the  same  angle.  The  following  rules 
take  in  all  the  possible  cases. 

The  reason  of  these  rules  will  be  made  apparent  by 
drawing  a  figure  to  represent  any  particular  case. 

1.  Deduct  the  bearing  of  the  side  that  is  to  be  made  a 
meridian  from  all  those  bearings  that  are  between  the  same 
points  as  it  is,  and  also  from  those  that  are  between  points 
directly  opposite  to   them.     If  it  is  greater  than   any  of 
those  bearings,  take  the  difference,  and  change  west  to  east, 
or  east  to  west. 

2.  Add  the  bearing  of  the  side  that  is  to  be  made  a 
meridian  to  those  bearings   that  are  neither  between  the 
same  points  as  it  is,  nor  between  points  directly  opposite. 
If  either  of  the  sums  exceeds  90°,  take  the  supplement,  and 
change  south  to  north,  or  north  to  south. 

EXAMPLES. 
Ex.  1.  The  bearings  of  a  tract  of  land  are,— 1.  K  57°  E.; 


212  COMPASS  SURVEYING.  [CHAP.  V. 

2.  K  89°  E.;  3.  S.  49J°  E.;  4.  South;  5.  S.  27f°  "W.;  6. 
S.  53|°  W.;  7.  F.  89°  W.;  8.  F.  37°  "W.;  9.  1ST.  43°  E.  to 
the  place  of  beginning.  Required  to  change  the  bearings, 
so  that  the  ninth  side  may  be  a  meridian. 

1.  F.  57°  E.  2.  F.  89°  E.  3.  S.  49J°  E. 

F.  43°  E.  F.  43°  E.  F.  43°    E. 

N.  14°  E.  F.  46°  E.  92J° 

180° 
F.  87J°  E. 

4.  S.  0°  W.    5.  S.  27}°  W.    6.  S.  53J°  W. 
K  43°  E.      K  43°  E.     1ST.  43  °  E. 
S.  43°  E.      S.  15i°  E.      S.  10J°  W. 

7.  K  89°  W.    8.  F.  37°  W.    9.  Forth. 
K  43°  E.      K  43°  E. 
132°        F.  80°  W. 
180° 
S.  48°  W. 


Ex.  2.  Change  the  bearings  in  the  following  notes,  so 

that  the  second  side  may  be  a  meridian  :  —  1.  IN".  43°  25'  W.  ; 

2.  K  29°  48r  E.  ;  3.  S.  80°  E.  ;  4.  K  89°  55r  E.  ;  5.  S.  10° 

13'  E.;  6.  K63°55'W.;  7.  S.63°45'W.;  8.  N.57°35'W. 

Ans.   1.  K  73°  13'  W.;    2.  North;    3.  K  70°  12'  E.; 

4.  K  60°  7'  E.  ;     5.  S.  40°  1'  E.;     6.  S.  86°  17'  W.;     7. 

5.  33°  57'  W.  ;   8.  K  87°  23'  W. 


Ex.  3.  Change  the  bearings  in  the  following  notes,  so 
that  the  fourth  side  may  be  a  meridian  :—  1.  S.  63°  E.  ;  2. 
8.  47°  E.;  3.  S.  59J°  W.;  4.  K  841°^.;  5.  K  12°  W.; 
6.  K  17J°  E.,  and  7.  S.  29|°  W. 

Ans.   S.  21J0  W.;   2.  S.  37i0"W".;   3.  K36i°"W.;  4. 
Forth;   5.  K  72J°  E.;   6.  S.  78°  E.;   7.  F.  65  j°  W. 


SEC.  VIII.  ]  SUPPLYING  OMISSIONS.  213 

SECTION  VIII. 
SUPPPLYING  OMISSIONS, 

349.  WHEN  any  two  of  the  dimensions  have  been  omit- 
ted to  be  taken,  or  have  become  obliterated  from  the  field- 
notes,  these  may  be  supplied.     This  should  never  lead  the 
surveyor  to  neglect  to  take  every  bearing  and  every  dis- 
tance.    It  is  far  better  to  use  almost  any  means,  however 
indirect,  to  obtain  all  the  bearings  and  distances  indepen- 
dently of  one  another  than  to  determine  any  one  from  the 
rest.     If  one  side  is  determined  from  the  others,  all  the 
errors  committed  in  the  measurements  are  accumulated  on 
that  side,  and  thus  the  means  of  proving  the  work  by  the 
balancing  of  the  latitudes  and  departures  is  lost.      The 
various  problems  in  Section  %  will  enable  the  young  sur- 
veyor to  solve  almost  every  case  of  difficulty  that  will  be 
likely  to  occur  in  making  his  measurements.     Should  any 
difficulty  arise  to  which  none  of  the  methods  there  de- 
veloped are  applicable,  a  knowledge  of  the  principles  of 
Trigonometry  will  afford  him  the  means  of  overcoming  it. 

CASE  1. 

350.  The  bearings  and  distances  of  all  the  sides  except 
one,  being  given,  to  determine  these. 

Determine  the  latitudes  and  departures  of  those  sides  of 
which  the  bearings  and  distances  are  given.  Take  the 
difference  between  the  sums  of  the  northings  and  southings, 
and  also  between  the  sums  of  the  eastings  and  westings : 
the  remainders  will  be  the  latitude  and  departure  of  the 
side  the  bearing  and  distance  of  which  are  unknown. 
With  this  latitude  and  departure  calculate  the  bearing  and 
distance  by  Art.  333. 

This  principle  will  enable  us  to  determine  a  side  when  it 
cannot  be  directly  measured.  Thus,  run  a  series  of  courses 
and  distances,  so  as  to  join  the  two  points  to  be  connected. 


214 


COMPASS  SURVEYING. 


[CHAP.  V. 


These,  with  the  unknown  side,  form,  a  closed  tract,  the 
sides  of  which  are  all  known  except  one. 

It  will  likewise  enable  us  to  determine  the  course  and 
distance  of  a  straight  road  between  two  points  already 
connected  by  a  crooked  one.  In  both  these  cases  it  is  best, 
where  the  nature  of  the  ground  will  admit  of  it,  to  run  the 
courses  at  right  angles  to  each  other,  as  in 
Fig.  153,  in  which  AB  is  the  distance  to 
be  determined.  Run  AC  any  direction, 
CD  perpendicular  to  AB,  DE  to  CD,  EF 
to  DE,  FG  to  EF,  and,  finally,  GB  per- 
pendicular to  FG  through  B. 

Then,  assuming  AC  as  a  meridian,  AC 
+  DE  +  FG  will  be  the  latitude  of  AB 
and  CD  +  EF  +  GB  the  departure.  From 
these  calculate  the  distance  AB  and  the 
bearing  BAC.  This  angle  applied  to  the 
true  bearing  of  AC  will  give  that  of  AB. 


EXAMPLES. 

Ex.  1.  The  bearings  and  distances  of  the  sides  of  a  tract 
of  land  being  as  follows,  it  is  desired  to  find  the  bearing 
and  distance  of  the  third  side,— viz. :  1.  K  56J°  "W.  15.35 
chains;  2.  K  9° W.  19.51  ch.;  3.  Unknown;  4.  S.  39|°  E. 
13.35  ch.;  5.  K  82J°  E.  12.65  ch.;  6.  S.  6j°  "W.  12.18  ch.; 
7.  S.  52J°  W.  20.95  ch. 


SEC.VIIL] 


SUPPLYING  OMISSIONS. 


215 


Sta. 

Bearing. 

Distance. 

N. 

s. 

E. 

w. 

1 

ET.  56i°  W. 

15.35 

8.53 

12.76 

2 

N.   9°    W. 

19.51 

19.2T 

3.05 

3 

4 

S.  39|°  E. 

13.35 

10.26 

8.54 

5 

K  82J°  E. 

12.65 

1.65 

12.54 

6 

S.    6£°  W. 

12.18 

12.10 

1.43 

7 

S.  52J°  W. 

20.95 

12.75 

16.62 

r-~ 

29.45 

35.11 
29.45 

21.08 

33.86 
21.08 

5.66  N. 


12.78  E. 


Diff.  Lat. 
Departure, 
Bearing, 

5.66 

12.78 
K  66°  7'  E. 

log.   0.752816 
log.   1.106531 

tang.  10.353715 

Bearing, 
Diff.  Lat. 
Distance, 


66°  T 


13.98 


cos.   9.607322 

log.   0.752816 

1.145494 


Ex.  2.  One  side  AB  of  a  tract  of  land  running  through  a 
swamp,  it  was  impossible  to  take  the  bearing  and  distance 
directly.  I  therefore  took  the  following  bearings  and  dis- 
tances on  the  fast  land, —viz. :  AC,  K  47°  W.  16.55  chains ; 
CD,  K  19°  5'  E.  11.48  ch. ;  DE,  K  11°  5'  W.  15.53  ch. ; 
EF,  K  23°  E.  9.72  ch.,  and  FB,  K  75°  12'  E.  14.00  chains. 
Required  the  bearing  and  distance  of  AB. 


216 


COMPASS  SURVEYING. 


[CHAP.  V. 


Sta. 

Bearing. 

Distance. 

N. 

s. 

E. 

w. 

A 

1ST.  47°  W. 

16.55  ( 

11.29 

12.10 

C 

K19°5'E. 

11.48 

10.85 

3.75 

D 

N.H05'W. 

15.53 

15.24 

2.99 

E 

1ST.  23°     E. 

9.72 

8.95 

3.80 

F 

1ST.  75°  12'  E. 

14.00 

3.58 

13.54 

B 

(49.91) 

(6.00) 

49.91 

21.09 

15.09 

Diff.  Lat. 
Departure, 
Bearing  AB, 

Bearing, 
Diff.  Lat. 
Distance, 


49.91 
6.00 
6°  51'  E. 

6°  51' 
50.27 


15.09 
6.00 

log.  1.698188 

log.  0.778151 

tang.  9.077963 

cos.  9.996889 
1.698188 
1.701299 


NOTE. — In  calculations  of  this  kind,  it  is  sufficiently  accurate  to  confine  the 
operations  to  two  decimal  places,  unless  the  number  of  sides  is  large.  In  Ex. 
2,  had  the  work  been  extended  to  the  third  decimal  place,  it  would  not  have 
made  more  than  15"  difference  in  the  bearing  and  1  link  in  the  distance. 

Ex.  3.  Given  the  bearings  and  distances  as  follows, — viz. : 
1.  S.  29f°  E.  3.19;  2.  S.37i°W.  5.86;  3.  S.39J°E.  11.29; 

4.  N.  53° E.  19.32;  5.  Unknown;  6.  S.  60£°  W.7.12;  7.  S. 
29J°E.  2.18;  8.S.  60|°  W.8.12;  to  find  the  bearing  and  dis- 
tance of  the  fifth  side.  Ans.  N".  31°  5'  W.  16.26  ch. 

Ex.  4.  Required  the  bearing  and  distance  of  the  third 
side  from  the  following  notes  :— 1.  K  46°  40' W.  18.41 
chains;  2.  K  54 J°  E.  13.45  chains;  3.  Unknown;  4.  S. 
74°  55'  E.  17.58  chains;  5.  S.  47°  50'  E.  15.86  chains ;  6. 

5.  47°  25' W.  16.36  chains;  7.  S.  62°  35' W.  14.69 chains. 

Ans.    3d  side,  K  5°  26'  W.  12.67  ch. 

Ex.  5.  It  being  impossible  to  take  the  bearing  and  dis- 
tance of  one  side  AB  of  a  tract  of  land  directly,  in  con- 


SEC. VIII.]  SUPPLYING  OMISSIONS.  217 

sequence  of  a  marsh  grown  up  with  thick  bushes,  I  took 
bearings  and  distances  on  the  fast  land  as  below, — viz.: 
AC  S.  49J°  W.  9.30  chains ;  CD  S.  32J0  E.  10.25 chains ;  DE 
S.  5J°  W.  6.75  chains ;  and  EB  K  79f  °  E.  8.10  chains.  Ke- 
quired  the  bearing  and  distance  of  the  side  AB. 

Ans.   S.  16°  12'  E.  20.82  ch. 

Ex.  6.  The  bearings  and  distances  taken  along  the  middle 
of  a  road  which  it  is  desired  to  straighten  are  as  below, — 
1.  S.  27°  30' E.  12.65  chains;  2.  S.  10J°  E.  23.45  chains;  3. 
S.  14°  W.  124.33 chains;  4.  S.  67°  E.  82.43 chains;  5.  S.  17° 
E.  96.35  chains.  Required  the  bearing  and  distance  of  a 
new  road  that  shall  connect  the  extremities. 

Ans.   S.  16°  44' E.  291.63  ch. 

CASE  2. 

351.  The  bearings  and  distances  of  the  sides  of  a  tract 
of  land  being  given,  except  two, — one  of  which  has  the 
bearing  given,  and  the  other  the  distance  and  the  points 
between  which  it  runs, — to  determine  the  unknown  bearing 
and  distance. 

RULE. 

Change  the  bearings  so  that  the  side  whose  bearing  only 
is  given,  may  be  a  meridian. 

Take  out  the  latitudes  and  departures  according  to  these 
changed  bearings.  Take  the  difference  of  the  eastings  and 
westings:  this  difference  will  be  the  departure  of  the  side 
not  made  a  meridian. 

"With  this  departure  and  the  given  distance,  calculate  by 
Art.  333  the  changed  bearing  and  difference  of  latitude, 
and  place  the  latter  in  the  column  of  latitude.  From  the 
changed  bearing  the  true  bearing  may  readily  be  found. 

Take  the  difference  between  the  northings  and  south- 
ings. This  difference  is  the  difference  of  latitude  of  the 
side  made  a  meridian,  and  is  equal  to  the  distance. 

NOTE. — In  general,  there  will  be  no  difficulty  in  determining  whether  the 
changed  bearing  found  should  be  north  or  south.  In  some  cases,  however, 
either  will  render  the  true  bearing  conformable  to  the  points  given.  In  this 
case  the  question  is  ambiguous,  and  can  only  be  determined  from  the  other 
data,  except  when  the  true  bearing  is  nearly  known. 


218 


COMPASS  SURVEYING. 


[CHAP.  Y. 


EXAMPLES. 

Ex.  1.  Given  the  courses  and  distances  as  below,  to  find 
the  unknown  bearing  and  distance. 


Sta. 

Bearing. 

Changed 
Bearing. 

Dist. 

N. 

s. 

E. 

w. 

1 

K  56J-  "W. 

S.  57f  W. 

15.35 

8.19 

12.98 

2 

K  9  W. 

K  75  W. 

19.51 

5.05 

18.85 

3 

K  66  E. 

North. 

(14.00) 

4 

S.  39f  E. 

K  74J  E. 

13.35 

3.62 

12.85 

5 

N.       .E. 

12.65 

(12.12) 

(3.62) 

6 

S.  6f  W. 

S.  59J  E. 

12.18 

6.23 

10.47 

7 

S.  52i  w. 

S.  131  E. 

20.95 

20.37 

4.89 

34.79 

34.79 

31.83 

31.83 

Dist.,  fifth  side, 
Dep.  « 

Ch.  bear.   « 


12.65 
3.62 
le^ 
66° 


A.  C.  8.897909 
0.558709 


sin.  9.456618 


K  82°  38'  E.,  bearing  of  fifth  side. 

Ch.  bear.,  fifth  side,  16°  38;  cos.  9.981436 

Dist.               «  1.102091 

Diff.  Lat.        «        .  12.12  "1.083527 

Dist,  third  side,  14.00  ch. 

Ex.  2.  Given—  1.  K  47°  W.  16.55  chains;  2.  N.  19°  5'  W. 
11.48  chains;  3.  K  -  W.  15.53  chains  ;  4.  K  23°  E.  9.72 
chains;  5.  !N".  75J°  E.  14  chains;  6.  S.  7°  E.,  unknown; 
to  determine  the  bearing  of  the  third  and  the  distance  of 
the  sixth  side. 

Ans.  3d  side,  K  28J°  W.  ;  6th,  48.67  ch. 


SEC.  VIII.] 


SUPPLYING  OMISSIONS. 


219 


CASE  3. 

352.  The  bearings  and  distances  of  the  sides  of  a  tract 
of  land  being  given,  except  the  distances  of  two  sides,  to 
determine  these. 


RULE. 

Change  the  bearings  so  that  one  of  the  sides  the  dis- 
tance of  which  is  unknown  may  be  a  meridian.  Take  out 
the  latitudes  and  departures  with  these  changed  bearings. 
The  difference  of  the  eastings  and  westings  will  be  the  de- 
parture of  the  side  not  made  a  meridian.  With  this  de- 
parture and  the  changed  bearing,  find  the  distance  and 
difference  of  latitude.  Place  the  latter  in  its  proper  place 
in  the  table.  Take  the  difference  between  the  northings 
and  southings:  this  difference  will  be  the  difference  of 
latitude  of  the  side  made  a  meridian,  and  will  be  equal  to 
the  distance. 

EXAMPLES. 

Given  as  follow,—!.  K  56J°  W.  15.35  chains ;  2.  K  9°  W., 
unknown ;  3.  K  66°  E.  14.00  chains ;  4.  S.  39f  °  E.  13.35 
chains;  5.  K  82}°  E.,  unknown ;  6.  S.  6}°"W.  12.18  chains; 
7.  S.  52J°  W.  20.95  chains;  to  find  the  distances  of  the 
second  and  fifth  sides. 


Sta. 

Bearing. 

Changed 
Bearing. 

Dist. 

N. 

S. 

E. 

w. 

1 

K56JW. 

K47JW. 

15.35 

10.42 

11.27 

2 

K  9  W. 

North. 

(19.54) 

(19.54) 

3 

N.  66  E. 

K  75  E. 

14.00 

3.62 

13.52 

4 

S.  39|  E. 

S.  30f  E. 

13.35 

11.47 

6.83 

5 

K82fE. 

S.  88J-  E. 

.39 

(12.64) 

6 

S.  6f  W. 

S.  15f  W. 

12.18 

11.72 

3.31 

7 

S.52JW. 

S.  61J  W. 

20.95 

10.00 

18.41 

33.58 

33.58 

32.99 

32.99 

220  COMPASS   SURVEYING.  [CHAP.  V. 

Ch.  bear.,  fifth  side,  88°  15'  A.  C.  sin.  0.000203 

Dep.                «         12.64  1.10174T 

Disk                "         12.65  1.101950 

Ch.  bear.  cos.  8.484848 

Bist.  1.101950 


Diff.  Lat.  0.39  S.  —  1.596798 

Ex.  2.  Given— 1.  S.  29|°  E.  3.19  chains;  2.  S.  37J°  W. 
5.86  chains;  3.  S.  39J°  E.,  unknown;  4.  K  53°  E.  19.32 
chains;  5.  K  31°  5'  W.,  unknown;  6.  S.  60f °  W.  7.12 
chains;  7.  S.29|°E.2.18  chains;  8.  S.  60J°  W.  8.12  chains; 
to  find  the  distances  of  the  third  and  fifth  sides. 

Ans.  3d  side,  11.28  chains;  5th,  16.26  chains. 

CASE  4. 

353.  The  bearings  and  distances  of  all  the  sides  of  a  tract 
of  land  being  known  except  the  bearings  of  two  sides,  to 
determine  these. 

EULE. 

Take  out  the  differences  of  latitude  and  the  departures 
of  the  sides  whose  bearings  and  distances  are  known.  The 
differences  of  the  northings  and  southings  will  be  the  dif- 
ference of  latitude,  and  that  of  the  eastings  and  westings 
the  departure,  of  a  line  which,  with  the  known  sides  of  the 
survey,  will  form  a  closed  figure,  and  may  therefore  be  called 
the  closing  line. 

With  this  closing  line  and  the  distances  of  the  two  other 
sides  form  a  triangle. 

Calculate  two  angles  of  this  triangle.  These  angles 
applied  to  the  bearing  of  the  closing  line  will  give  the 
bearings  required. 


SEC.  VIII.] 


SUPPLYING  OMISSIONS. 


221 


EXAMPLES. 

Ex.  1.  Given  AB  (Fig.  154)  K  56 J°  W.  15.35  chains;  BC 
K  9°  W.  19.51  chains ;  CD  K  —  E.  14  chains ;  DE  S.  39f° 
E.  13.35;  EF  F.  82J°  E.  12.65  chains;  FG  S.  —  W.  12.18 
chains;  GA  S.  52J°  W.  20.95  chains;  to  find  the  bearings 
of  the  third  and  sixth  sides. 


Bearing. 

Disk 

N. 

S. 

E. 

w. 

AB 

N.  56J  W. 

15.35 

8.53 

12.76 

BC 

K  9  W. 

19.51 

19.2T 

3.05 

Ce 

S.  39|  E. 

13.35 

10.26 

8.54 

ef 

K  82J  E. 

12.65 

1.65 

12.54 

GA 

S.  52J  W. 

20.95 

12.75 

16.62 

29.45 
23.01 

23.01 

21.08 

32.43 

21.08 

6.44 

11.35 

Diff.  Lat. 

Dep. 

Tang,  closing  line, 

Cos.  bear. 

Diff.  Lat. 

Dist.  closing  line, 

FG 

/G 
/F 


6.44 
11.35 

S.60°26'E. 

60°  26' 
13.05 


A.  C.  9.191114 

1.054996 

10.246110 

A.C.  0.306769 

0.808886 
1.115655 


A.  C.  8.884388 

"  "  8.853872 

1.292588 
0.871281 


JF/G 
F/G 


26°  41' 

53°  22' 


2)19.902129 
cos.  9.951064 


222  COMPASS  SURVEYING.  [CHAP.  V. 

FG  12.18  A.  C.  8.914353 

/F  14.00  1.146128 

sin.  F/G  53°  22'  9.904429 

sin./GF  67°  17'  9.964910 

60°  26' Bear,  of /G 
S.    6°51'W.     "     GF 
180°  -  (53°  22'  +  60°  26')  =  66°  12'; 
therefore,    K  66°  12'  E.  is  the  bearing  of  CD. 

Ex.  2.  Given— 1.  S.  29f°  E.  3.19  chains;  2.  S.  37J°  W. 
5.86  chains ;  3.  S.  —  E.  11.29  chains ;  4.  BT.  53°  E.  19.32 
chains;  5.  K  —  W.  16.26  chains;  6.  S.60f°W.  7.12 chains; 
7.  S.  29J°  E.  2.18'chains;  8.  S.  60|°  W.  8.12  chains;  to 
find  the  bearing  of  the  third  and  fifth  sides. 

Ans.   3d  side,  S.  39°  8'E.;  5th,  K  31°  W. 

354.  The  first  three  of  the  preceding  rules  are  so  simple 
as  hardly  to  need  any  explanation.  The  principle  of  the 
last  will  be  seen  from  the  following  illustration.  The  figure 
being  protracted  from  the  field-notes  in  Ex  1,  Case  4,  these 
are,  as  will  be  seen,  the  same  as  Ex.  1  in  the  other 
cases. 

Let  ABCDEFG  (Fig.  154)  be  the 
plat  of  the  tract,  the  bearings  of  CD 
and  FG  being  supposed  unknown. 
If  Ce  and  ef  be  drawn  parallel  to 
the  sides  DE  and  EF,  and  /G  be 
joined,  then  will  ABCe/G  form  a 
closed  figure,  the  bearings  and  dis- 
tances of  all  the  sides  except  /G 
being  known.  The  course  and  dis- 
tance of  this  side,  which  is  the  closing  line,  are  found  as 
directed  in  the  rule.  Join  /F  and  eE.  Then  /F  is  equal 
and  parallel  to  eE  and  therefore  to  CD.  The  sides  of  the 
triangle  /FG  are  therefore  the  closing  line,  the  side  FG, 
and  the  line  /F  equal  and  parallel  to  the  side  CD.  In/FG 
find  the  angles /and  G:  these  applied  to  the  bearing  of /G 
will  give  the  bearings  of  /F  or  CD  and  of  FG. 


SEC.  IX.]  CONTENT  OF  LAND.  223 

This  method  might  have  been  employed  in  Cases  2  and  3. 
Those  given  in  the  rules  are,  however,  more  concise,  and 
are  therefore  to  be  preferred. 

355.  Though  the  methods  illustrated  above  will  serve  to 
supply  omissions  in  all  cases  where  not  more  than  two  of 
the  dimensions  are  unknown,  yet  it  will  not  be  amiss  again 
to  impress  on  the  young  practitioner  the  necessity,  in  all 
cases  in  which  it  is  practicable,  of  determining  each  side 
independently  of  every  other.  The  rules  for  supplying 
omissions  should  only  be  used  in  cases  where  one  or  more 
of  the  data  have  been  accidentally  omitted,  or  have  become 
defaced  on  the  notes.  However  accurate  the  field-work  may 
be,  there  is  always  a  liability  to  error,  and  if  one  side  is 
determined  by  the  rest  no  means  are  left  of  detecting  any 
error.  When  a  side  cannot  be  measured  directly,  the  best 
way  is  to  determine  it  by  some  of  the  trigonometrical 
methods,  taking  the  angles  and  base-lines  with  great  care. 
In  this  way  a  degree  of  accuracy  may  be  obtained  equal  to 
that  of  the  sides  measured  directly.  The  latitudes  and  de- 
partures may  then  be  balanced  as  usual. 


SECTION  IX. 

CONTENT  OF  LAND. 

356.  FROM  the  bearings  and  distances  of  the  sides  of  a 
tract  of  land,  or  from  the  angles  arid  the  lengths  of  the 
sides,  the  area  may  be  found,  however  numerous  the  sides 
may  be.  This  may  be  done  by  Problem  4,  which  is  entirely 
general,  it  being  applicable  whatever  the  number  of  sides 
may  be,  provided  they  are  straight  lines.  As,  however, 
there  are  other  more  concise  methods  applicable  to  triangles 
and  quadrilaterals,  those  are  first  given. 

If  one  or  more  of  the  boundaries  is  irregular,  instead  of 
multiplying  the  number  of  sides  by  taking  the  bearings  of 


224  COMPASS  SURVEYING.  [CHAP.  V. 

all  the  sinuosities  of  tlie  boundary,  it  is  better  to  run  one  or 
more  base  lines  and  take  offsets,  as  directed  in  chain  sur- 
veying. The  content  within  the  base  lines  is  then  to  be 
calculated,  and  the  area  cut  off  by  the  base  lines,  being 
found  by  the  method  Art.  256,  is  to  be  added  to  or  sub- 
tracted from  the  former  area,  according  as  the  boundary  is 
without  or  within  the  base. 

As  has  been  already  remarked,  (Art.  257,)  when  the  tract 
bounds  on  a  brook  or  rivulet,  the  middle  of  the  stream  is 
the  boundary,  unless  otherwise  declared  in  the  deed.  Lands 
bordering  on  tide  water  go  to  low-water  mark.  "When*  the 
stream,  though  not  tide  water,  is  large,  the  area  is  generally 
limited  by  the  low-water  mark,  or  by  the  regular  banks  of 
the  stream. 

If  the  farm  bounds  on  a  public  road,  the  boundary  is, 
except  in  special  cases,  the  middle  of  the  road,  and  the 
measures  are  to  be  taken  accordingly. 

357.  Problem  1. — Given  two  sides  and  the  included  angle 
of  a  triangle  or  parallelogram,  to  determine  the  area. 

Say,  As  radius  is  to  the  sine  of  the  included  angle,  so  is 
the  rectangle  of  the  given  sides  to  double  the  area  of  the  tri- 
angle, or  to  the  area  of  the  parallelogram. 

DEMONSTRATION. — We    have,    (Fig.  155,)   by 
Art.  137,— 

Asrad.  :  sin.  A  : :  AC  :  CD  :  :  AB.AC  :  AB.CP 
CD,  (Cor.  1.6) ;  but  AB  .  CD  =  2  ABC. 

EXAMPLES. 

Ex.  1.   Given  AB  =  12.36  chains, 
BC  =  14.36  chains,  and  ABC  =  47°  35',  to  determine  the 
area  of  the  triangle. 

As  rad.  A.C.  0.000000 

:    sin.  B  47°  35'  9.868209       . 

f  AB  12.36  ch.  1.092018 

:  I  BC  14.36  1.157154 

:    2  ABC  2)131.033  2.117381 

65.5165  ch.  =  6  A.,  2  K.,  8.26  P. 


SEC.  IX.]  CONTENT  OF  LAND.  225 

Ex.  2.  Given  AB  K  37°  14'  W.  1T.25  chains,  and  BC 
K  74°  29'  W.  10.8T  chains,  to  determine  the  area  of  the 
triangle  ABC.  Ans.  5  A.,  2  R,,  28  P. 

Ex.  3.  Given  AB  =  23.56  chains,  AC  =  16.42  chains,  and 
the  angle  A  126°  47'.  Required  the  area  of  the  triangle. 

Ans.   15  A.,  1  R,,  38.7  P. 

358.  Problem  2. — The  angles  and  one  side  of  a  triangle 
being  given,  to  determine  the  area. 

Say,  As  the  rectangle  of  radius  and  sine  of  the  angle  op- 
posite the  given  side  is  to  the  rectangle  of  the  sines  of  the 
other  angles,  so  is  the  square  of  the  given  side  to  double 
the  area. 

DEMONSTRATION. — We  have  (Fig.  155) 

r  :  sin.  A  : :  AC  :  CD  (Art.  137), 
and  Bin.  B  :  sin.  C  : :  AC  :  AB  (Art.  139). 

(23.6).  r .  sin.  B  :  sin.  A.  sin.  C  : :  AC2 :  AB .  CD,  or  2  ABC. 

EXAMPLES. 

Ex.  1.  Given  AB  =  21.62  chains,  and  the  angle  A=  47° 
56'  and  B  =  76°  15',  to  find  the  area. 

frad.  A.C.  0.000000 

8  (sin.  C  55°  49'  «    0.082366 

fsin.  A         47°  56'       9.870618 

I  sin.  B         76°  15'       9.987372 

JAB  21.62  ch.  1.334856 

1  :    t AB  21.62  1.334856 

:       2  ABC  2)407.444  2.610068 

Area  =  203.722  ch.  =  20  A.,  1  R.,  19.5  P. 


Ex.  2.  Given  AB  17.63  chains,  and  the  angle  A  =  63° 
52'  and  B  73°  47',  to  find  the  area. 

Ans.   19  A.,  3  R.,  22  P. 

Ex.  3.  Given  one  side  15.65  chains,  and  the  adjacent 
angles  63°  17'  and  59°  12',  to  determine  the  area  of  the 
triangle.  Ans.  11  A.,  0  R.,  22  P. 

15 


226 


COMPASS   SURVEYING. 


[CHAP.  V. 


359.   Problem  3. — To  determine  the  area  of  a  trapezium, 
three  sides  and  the  two  included  angles  being  given. 

KULE. 

1.  Consider  two  adjacent  sides  and  their  contained  angle 
as  the  sides  and  included  angle  of  a  triangle,  and  find  its 
double  area  by  Prob.  1. 

2.  In  like  manner,  find  the  double  area  of  a  triangle  of 
which  the  two  other  adjacent  sides  and  their   contained 
angle  are  two  sides  and  the  included  angle. 

3.  Take  the  difference  between  the  sum  of  the  given 
angles  and  180°,  and  consider  the  two  opposite  given  sides 
and  this  difference  as  two  sides  and  the  included  angle  of  a 
triangle,  and  find  its  double  area. 

4.  If  the  sum  of  the  given  angles  is  greater  than  180°, 
add  this  third  area  to  the  sum  of  the  others ;   but  if  the 
sum  of  the  given  angles  is  less  than  180°,  subtract  the  third 
area  from  the  sum  of  the  others:  the  result  will  be  double 
the  area  of  the  trapezium. 


DEMONSTRATION. — Let  ABCD  (Figs.  156,  157)  be 
the  trapezium,  of  which  AB,  BC,  and  CD,  and  the 
angles  B  and  C,  are  given. 

Join  BD,  and  draw  DE  and  CG  perpendiculai\to 
AB,  and  CF  perpendicular  to  ED.     Then  will  DC5\ 
=  180°  oo  (B  -f  C.)     Also,  draw  AH  parallel  to  \G 
CB,  and  join  DH. 

Then  will  2  ABD  ==  AB  .  DE  =  AB  (EF  db  DF) 
=  AB.EF±AB.DF  =  2  ABC  ±  2  CDH. 


Whence  2  ABCD  =  2  BDC  +  2  ADB  =  2  BCD  -f 
2  ABC  ±2  CDH:  the  plus  sign  being  used  (Fig. 
157)  when  the  sum  of  the  angles  is  greater  than 
180°. 


Fig.  156. 


SEC.  IX.] 


CONTENT  OF  LAND. 


227 


EXAMPLES. 

Ex.  1.  Given  AB  =  6.95  chains,  BC  =  8.37  chains,  CD 
=  5.43  chains,  ABC  =  85°  17',  and  BCD  =  54°  12',  to  find 
the  area  of-  the  trapezium. 

As  r  0.000000 

:     sin.  B                        85°  17'  9.998527 

f  AB                            6.95  0.841985 

1  :  I  BC                            8.37  0.922725 

:    2ABC 


57.975 


As  r 

:     sin.  180°  -  (B  +  C)  40°  31' 
AB  6.95 

CD  5.43 

2  CDH  25.031 


1.763237 

0.000000 
9.812692 
0.841985 
0.743800 
1.398477 


As  r 
:     sin.  C 
(BC 

:  [CD 

:    2  BCD 


0.000000 
9.909055 
0.922725 
0.734800 
1.566580 


34.903ch.  =  3  A.,  IE.,  38.45R 

Ex.  2.  Given  AB  S.  27°  E.  12.47  chains,  BC  K.  66°  E. 
11.43,  and  CD  K  8°  "W.  9.16  chains,  to  find  the  area  of 
the  trapezium.  Ans.  14  A.,  0  R,  1.56  P. 

Ex.  3.  Given  AB  S.  45°  W.  8.63  chains,  BC  S.  86° 
30'  E.  9.27  chains,  and  CD  K  34°  E.  11.23  chains,  to  find 
the  area  of  the  trapezium. 

Ans.   6  A.,  2  R.,  9  R 


228 


COMPASS   SURVEYING. 


[CHAP.  V 


360,  The  above  rule  is  a  particular  example  of  a  more 
general  problem,  which  may  be  enunciated  thus : — 

Let  A,  B,  C,  D,  &c.  be  the  sides  of  any  polygon.,  and  let 
the  angle  contained  between  the  directions  of  any  two 
sides,  as  B  and  D,  be  designated  [BD].  Then,  leaving  out 
any  side,  we  shall  have  the  double  area  equal  to  the  sum 
of  the  products  of  all  the  other  pairs  into  the  sine  of  their 
included  angle.  Thus,  if  the  figure  were  a  pentagon,  we 
should  have  2  the  area  =  BC  sin.  [BC]  +  BD  sin.  [BD]  + 
BE  sin.  [BE]  +  CD  sin.  [CD]  +  CE  sin.  [CE]  +  DE  sin. 
[DE]. 

Observing  that  any  product  must  be  taken  negative,  if 
the  angle  is  turned  in  a  contrary  direction  from  the  general 
convexity  of  the  figure  with  reference  to  the  side  A. 

Thus,  in  Fig.  156,  we  have  2  ABCD  =  AB .  BC  sin. 
[AB  .  BC]  +  BC .  CD  sin.  [BC .  CD]  -  AB  .  CD  sin.  [AB . 
CD],  the  lines  BA  and  CD  meeting  so  as  to  make  the 
angle  [AB  .  CD]  present  its  convexity  in  the  opposite 
direction  from  that  of  the  figure. 

But,  in  Fig.  157,  we  have  2  ABCD  =  AB .  BC  sin. 
[AB.BC]  +  BC.CD  sin.  [BC.CD]  +  AB .  CD  sin. 
[AB.CD]. 

In  the  pentagon  (Fig.  158)  we  shall 
have 

2  Area  =  B.C.sin.[B.C.]  +  B.D.sin. 
[B.D.]  +  B.E.sin.[B.E.]-f  C.D.sin. 
[C.D.]+C.E.sin.[C.E.]  +  D.E.sin. 

[D.E]. 

In  Fig.  159  we  have 

2  Area  =  B.C.  sin.  [B.C.]+  B.D.sin. 
[B.D.]- B.E.sin.  [B.E.]+  C.D.sin. 
[C.D.]  +  C.E.sin.  fC.E.l  +  D.E.sin. 
[D.E]. 


Fig.  158. 
A 


Fig.  159. 


SEC.  IX.] 


CONTENT  OP  LAND. 


229 


361,  Problem  4. —  The  bearings  and  distances  of  the  boun- 
daries of  a  tract  of  land  being  given,  to  determine  its  area  by 
means  of  the  latitudes  and  departures  of  the  sides. 

Let  ABCDEFG-  (Fig.  160)  Fig.  ieo. 

be  the  plat  of  a  tract,  and  let  N 
MS  be  a  meridian  anywhere   a-- 
on  the  map.      Through   the 
corners  draw  the  perpendicu- 
lars Aa,  B6,  &c.    Then,  it  is  evi- 
dent that  ABCDEFG  =  AagGc 
+  DdeE  —  Aa&B  — 
-  CcdD  -  Ee/F. 

Now,  these  various  figures 
being  trapezoids,  their  areas   aL 
will  be  found  by  multiplying 
their    perpendiculars    by  the   g 
half-sums  of  their  parallel  sides. 

The  perpendiculars  are  the  differences  of  latitude  of  the 
sides  of  the  tract.  The  sums  of  their  parallel  sides  may 
be  found  as  follows : — 

The  position  of  the  line  ETS  being  arbitrary,  the  sum  Aa 
+  B6,  corresponding  to  the  first  side  AB,  may  be  taken  at 
pleasure.  Now,  if  from  Aa  +  B6  we  take  AA,  the  whole 
departure  of  the  two  sides  AB  and  BC,  we  have  B6  4-  Cc, 
the  sum  of  the  parallel  sides  of  B6cC.  Similarly,  if  to 
B£  -f  Cc  we  add  iD,  the  departure  of  the  two  sides  BC  and 
CD,  we  have  Cc  +  Dd;  and  so  on.  The  whole  may  be 
arranged  in  a  tabular  form,  as  below, — 


Sides. 

N. 

S. 

E. 

W. 

E.  D.  D. 

W.  D.  D. 

Multipliers. 

N.  Areas. 

8.  Areas. 

AB 

Bfc 

AA; 

Afc  +  Go 

Aa+B6,E. 

2  Aa6B 

BC 

fO 

Bp 

Afc  +  B^> 

B6  +  Cc,  E. 

2B6cC 

CD 

Cq 

qD 

gD  —  Bp 

Cc  +  Dd,  E. 

2  CcdD 

DE 

Dl 

ffi 

qD  +  lE 

Dd+Ee,  E. 

2DefeE 

EF 

~Em 

mF 

ZE  +  mF 

Ee+F/,  E. 

2Ee/F 

FG 

nG 

Fn 

mF  —  Fra 

P/+  Gsr,  E. 

2  FfgG 

QA 

oA 

Go 

Fn  +  Go 

G0  +  Aa.E. 

ZQgaA. 

in  which  the  first  column  contains  the  sides,  and  the  next 
four  the  differences  of  latitude  and  the  departures;   the 


230  COMPASS  SURVEYING.  [CHAP.  V. 

fifth  and  sixth  columns  contain  the  whole  departures  of 
two  consecutive  sides.  These  may  be  called  the  double 
departures,  and  the  columns  headed,  accordingly,  E.D.D. 
and  "W.D.D.  These  double  departures  are  found  thus: 
The  first,  AA;  4-  Go,  is  the  sum  of  the  departures  of  GA  and 
AB,  and  is  placed  in  the  column  of  west  double  departures, 
because  both  departures  are  westerly ;  the  second,  AA;  -f  B£>, 
is  the  sum  of  those  of  AB  and  BC,  and  is  west;  the  third 
is  Dq  —  Bp,  and  is  east,  because  D  is  east  of  B  ;  the  fourth, 
Dq  +  E£,  is  east ;  and  so  on.  The  eighth  column  contains 
the  sums  of  the  parallel  sides.  These  may  be  called  the 
multipliers.  They  are  found  by  the  following  process. 
Assuming  the  first,  Aa  +  B6,  at  pleasure,  designate  it 
either  east  or  west.  In  the  figure,  the  line  MS  being  to 
the  west  of  AB,  the  multiplier  is  east.  The  double  de- 
parture AA;  +  Dp  =  Ah  being  west,  subtract  it  from  Aa  -f 
B6,  and  we  have  Db  +  Cc.  To  B6  +  Cc  add  the  next 
double  departure,  qD  —  pE  =  £D,  and  we  have  Cc  +  Dd ; 
qD  +  ZE  added  to  Cc  +  Dd  gives  Dd  +  fte;  IE  +  mF  added 
to  Dd  +  Ee  gives  Ee  -f  F/ ;  mF  —  Fn  added  to  Ee  +  F/  gives 
F/  +  Gg ;  and,  lastly,  F/i  +  Go  taken  from  F/  +  Gg  leaves 
Gg  +  Aa. 

The  areas  are  arranged  in  the  last  two  columns,  which 
are  headed  north  areas  and  south  areas  for  distinction. 
These  areas  are  placed  in  the  above  table  in  the  columns 
of  the  same  name  as  the  difference  of  latitudes  of  the  sides 
to  which  they  belong. 

Had  the  line  NS  been  drawn  so  Eis-  1^J- 

as  to  intersect  the  plat,  some  of  the 
areas  would  have  been  to  the  west 
of  it,  and  some  of  the  multipliers 
might  have  been  west.  Fig.  161  is 
an  example  of  this. 

In  this  case,  we  have  * 
2  ABCDEFG  =  2  AafrB  +  2  DbcC 
-f  2  CcdD  —  2Ddr  +  2reE  —  2Ee/F 
+  2  F/^G  +  2  Ggs  -  2  saA  =  2 
AabD  +  2  DbcC  +  2  CcdD  -  2  (Ddr  ;s 

-  reE)  -  2  Ee/F  +  2  F/#G  +  2  (Ggs  -  saA.) 


SEC.  IX.] 


CONTENT   OF   LAND. 


231 


But 2(Ddr -  reE)  =  Dd .  dr  —  Ee  .  er  =  Dd .  de  -  Dd .  cr  - 
Ee.de  +  Ee.dr; 

and  since       Dd  :  dr  : :  Ee  :  er,  Dd.er  =  Ee  dr. 

2  (Ddr  -  reE)  =  Dd  .  de  -  Ee  .  de  =  (Dd  -  Ee)  de. 

Whence  2  ABCDEFG  =  (Aa  +  Bb)  ab  +  (Bb  +  Cc)  be  + 
(Cc  +  Dd)  cd  -  (Dd  -  Ee)  de  -  (Ee  -f /F)  e/  +  (/F  + 
+  (Gg  —  Aa)  ag. 

The  following  table  exhibits  the  whole. 


Sides. 

N. 

~I£ 

S. 

E. 

W. 

E.  D.  D. 

W.  D.  D. 

Multipliers. 

—  r 
N.  Areas. 

S.  Areas. 

AB 

pB 

jpB+Go 

B6  +  Aa,  W. 

2AafcB 

BC 

B? 

?c 

pB  +  qC 

B&-J-CC,  W. 

2B6Cc 

CD 

Di 

Gi 

Gi  —  qG 

Cc  +  Dd,  W. 

2Cafl> 

DE 

Et 

J)t 

Ci  +  Vt 

Dd  —  Ee,  W. 

2(Ddr  —  Eer) 

EF 
~FOT 

Em 

mV 

Dt  +  ¥m 

Ee+F/,  E. 

2(Ee/F) 

Gn 
~Ao" 



Fn 

"oT" 

Fw  —  Fra 

F/+G<7,E. 

2F/flrG 

GA 

F»  +  Go 

G^r  —  Aa,  E. 

2(Gps  —  Aas) 

Here  the  first  multiplier  is  west,  the  meridian  being  to 
the  east  of  the  line  AB.  The  subsequent  multipliers  are 
found  as  follow:— (Bb  +  Aa)  -f  (^B  +  #0)  =  Bb  +  Cc; 
(Bb  +  Cc)  -  (Ci  -  qC)  =  Cc  +  Dd;  (Cc  +  Dd)  -  (Of  +  D*) 
=  Dd  -  Ee ;  (D*  +  Fw)  -  (Dd  -  Ee)  =  (Ee  +  F/),  which 
must  be  marked  east,  not  only  from  its  position  on  the 
figure,  but  also  from  the  fact  that  the  east  double  departure 
is  greater  than  the  west  multiplier,  which  is  taken  from  it ; — 
(Ee  +  F/)  4-  (Fw  —  Fn)  =  F/+  G#;  and  (F/  -f  G#)  —  (Fn 
-f  Go)  =  Gg  —  Aa. 

The  areas  are  arranged  so  that  the  additive  quantities 
may  be  in  the  column  of  south  areas  and  the  subtractive 
in  that  of  north  areas. 

From  the  above  investigation  the  following  rule  is  de- 
rived : — 

EULE. 

Kule  a  table  as  in  the  adjoining  examples.  Find  the  cor- 
rected latitudes  and  departures  by  Art.  338.  Then,  if  the 
departures  of  the  first  and  last  sides  are  of  the  same  name, 
add  them  together,  and  place  their  sum  opposite  the  first 
side  in  the  column  of  double  departures  of  that  name ;  but 


232  COMPASS    SURVEYING.  [CHAP.  V. 

if  they  are  of  different  names,  take  their  difference  and 
place  it  in  the  column  of  the  same  name  as  the  greater. 
Proceed  in  the  same  way  with  the  departures  of  the  first 
and  second  sides,  placing  the  result  opposite  the  second 
side ;  and  so  on. 

Assume  any  number  for  a  multiplier  for  the  first  side, 
marking  it  E.  for  east  or  W.  for  west,  as  may  be  preferred. 
Then,  if  this  multiplier  and  the  double  departure  corre- 
sponding to  the  second  side  are  of  the  same  name,  add 
them  together,  and  place  the  sum  with  that  name  in  the 
column  of  multipliers,  for  a  multiplier  for  that  side ;  but, 
if  the  multiplier  and  double  departure  be  of  different 
names,  take  their  difference  and  mark  it  with  the  name 
of  the  greater,  for  the  next  multiplier.  Proceed  in  the 
same  manner  with  the  multiplier  thus  determined  and  the 
third  double  departure,  to  find  the  multiplier  for  the  third 
side.  So  continue  until  all  the  multipliers  have  been  found. 

Multiply  the  difference  of  latitude  of  each  side  by  the 
corresponding  multiplier,  for  the  area  corresponding  to 
that  side.  If  the  multiplier  be  east,  place  the  product  in 
the  column  of  areas  which  is  of  the  same  name  as  the  dif- 
ference of  latitude ;  but,  if  the  multiplier  be  west,  place 
the  product  in  the  column  of  the  opposite  name. 

Sum  the  north  and  the  south  areas.  Half  the  difference 
of  the  sums  will  be  the  area  of  the  tract. 

NOTE. — In  working  any  area,  the  columns  of  double  departures  should 
balance. 

The  first  multiplier  is  generally  assumed  zero.  One  multiplication  is  thus 
avoided.  When  this  is  done,  the  last  multiplier  will  be  equal  to  the  first  double 
departure,  but  of  a  different  name. 

EXAMPLES. 

Ex.  1.  Given  the  bearings  and  distances  as  follow,  to  find 
the  area:— 1.  K  56J°  W.  15.35  ch. ;  2.  N.  9°  W.  19.51  ch. ; 
3.  K  66°  E.  14.01  ch. ;  4.  S.  39J °  E.  13.35  ch. ;  5.  K  82i°  E. 
12.65  ch. ;  6.  S.  6f°  W.  12.18  ch. ;  7.  S.  52J°  W.  20.95  ch. ; 
to  find  the  area. 


IX.] 


CONTENT  OF  LAND. 


233 


I 

CO 
OS 
CD 

| 

§ 

CD 

0 
xO 

o 

rH 

0 

si 

CD   CO 
1C   OJ 
O  ^O 

5 

02 

£ 
CO 

CO 

CO 

XO 

xO 

«-0 

xO  OS 

iO   <M 

§  § 

3 

0 

rH 

CD 
OS 

OS 

0 
rH 
CO 
OS 

os  • 

Multipliers. 

I 
g 

15.82  W. 

g 

CO 

CO 
rH 

CO 
CO 
CO 

H 

xO 

HH 
OS 
CO 
OS* 

( 
i 

W.D.D. 

OS 
CO 

OS 

CO 

xd 

rH 

§ 

00 

CO 

CO 

i 
f 

ft 
ft 

a 

OS 

CO 
rH 

o 

rH 

rH 

S5 
s 

T 

c 

* 

CO 

cq 

rH 

g 
CO 

CO 
rH 

CO 
CO 
CD 
rH 

CO 
CO 

CO 
CO 

- 

o 

CO 
rH 

XO 
00 

i 
rH 

CO 
CO 

si 

02 

0 
rH 

0 

O 
rH 

<N 

i 

; 

CO 

CO 

OS 

rH 

OS 
CO 

CO 
rH 

(M 

rH 

xd 

CO 

j& 

O 

0 

o 

8=5 

rH 
O 

s 

rH 

O 

rH 
0 

o   - 

J 

0 

i 

CO 

CO* 

CO 
rH 

40 

rH 

CO     * 
CO 

co" 

1 

S 

q 

rf 

i—  1 

CO 

rH 

co  co  c 

CO  CO  < 

| 

02 

CO 

0 
rH 

0 
rH 

rH 

O 

rH 

xd 

CO 

M 

CO 
XO 
CO 

OS 

0 

xd 

® 

rH 

XO  rH   -. 

rH  rH   < 

§§' 

*< 

5 

j 

CO 

rH 

0 

CO 

CO 

CO 
rH 

C5 

§   ! 

H 
D 

ft 

XO 

OS 

M< 

CO 

C<1 

<N 

O 

CO     C 

Q 

^^ 

s 

Bearings. 

O 
xO 

os 

CO 
CO 

OQ 

CO 

CO 

od 

xO 
02 

1 

2 

H 
^ 

^ 

; 

co 

j 

XO 

CO 

*- 

3 


234 


COMPASS   SUKVEYLNG. 


[CHAP. 


Ex.  2.   Given  the  bearings  and  distances  as  in  the  ad 
joining  table,  to  calculate  the  area. 


0  0 

t 

I 

CO 
OS 
OS 
CO 

OS 
CO 

o 

1 

o 
o 

§ 

OS 
OS 

o 

CO 

tO 

CO 

O  ^  CO  CO  rfi 

CN  OS   CO   i—* 

rH  <M  CO  OS 

co    S 

•< 
od 

§ 

(N 

CO 

CO 

OS 

0 
CO 

CO 

0 

CO 

s 

t^  rH   tO   (N 
(N  <M  O   _•« 
CO     CO  O 

;=v  co 

^     CO 
rH 

fc 

53 

<M 

•1 

8 

w 

CO 

rH 

CO 

1—  1 

05 

o 

CO 

CO 

0 
rH 

CO 
tO 

E 

co 

rH 

3 
ft 

§ 

CO 
I—  1 

rH 

T* 

CO 

£ 

CO 

rH 

OS 

*• 

co 

p 

1 

rH 

tO 

CO 

o 

CO 

T—  1 

OS 
CO 

«    *1 

p 
p 

CO 
CO 

tO 
CO 
CO 

tO 
CM 
CO 

(N 

3 

OS 
CO 

2 

J5     « 

»o*       1~l 

* 

s 

CO 

CO 

CO 
cd 

OS 

o 

tO 

tO 
GO 

co        0 
to        co 

H 

CO 

OS 

CO 

CD 

3 

»o 

OS 

CO 

• 

<M 

S    -5 

02 

1 

rH 

rH 
0 

OS 

<M 

id 

CO 

CO 
OS 

OS 

»o 

CO 

o 

tO 
rH 

0 

» 

(N 
<M 

6^ 

o 

o 

O 

o 

0 

o   „ 

J* 

i 

rH 
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s 

o 

_„     g 
°.    ^ 

^ 

IO 
CO 

« 

CO 

OS 

s 

i 

OS        fn 

3  § 

CM   pq 

ri 

OS 
OS 
CO 

i 

CO 

s 

s 

OS 
CO 

•^  os  to 
CO  oq  0 

od 

tO 

CO 

to 
rH 

« 

tOrH|^ 

****•'«; 

^ 

s 

w 

CO 
CO 

OS 

OS 

CO 

10 

rH 

0 
rH 

tO 

OS 

-H      ^ 

S      0 

c^   5^ 

1 

<M 

CO 

CO 

§ 

OS 
(M 

o 

OS 

co 

CO 

co 

§ 

8 

CO 
CO 

CO 
rH 

•    S 

p 

*• 

CO 

o 

os 

T—  1 

to 

rH 

to 

rH 

(N 

<N 

w 

w 

£ 

^ 

£ 

w 

& 

H 

£ 

H 

£ 

1 

b 

co 

1 

& 

OS 

02' 

OS 
tO 

02 

0 

tO 
02 

to 

I 

Os 
o 

CO 

iM 

o 

tO 

tO 

o 

CO 
CO 

tO 
CO 

o 

CO 

CO 

?3 

1 

: 

« 

co 

•« 

o 

CO 

*• 

CO 

OS 

0 
rH 

r^ 

f 

SEC.  IX.]  CONTENT  OF  LAND.  235 

Ex.  3.  Given  the  bearings  and  distances  as  follow,  to 
calculate  the  area:— 1.  K  27°  15'  E.  7.75  ch.;  2.  S.  62°  25' 
E.  10.80  ch.;  3.  S.  7°  55'  E.  9.50  ch.;  4.  S.  47°  25'  E.  9.37 
ch. ;  5.  S.  54°  25'  W.  8.42  ch.;  6.  K  37°  35'  W.  23.69  ch. 

Ans.  22  A.,  1  E.,  26.17  P. 

Ex.  4.  Calculate  the  area  from  the  following  notes: — 

1.  K  46°  40'  W.  18.41  ch. ;  3.  N.  54°  30'  E.  13.45  ch. ;  3.  K 
5°  30'  W.  12.65  ch.:  4.  S.  74°  55'  E.  17.58  ch;   5.  S.  47° 
50'  E.  15.86  ch. ;   6.  S.  47°  25'  W.  16.36  ch. ;   7.  S.  62°  35' 
W.  14.69  ch. 

Area,  66  A.,  2  E.,  21  P. 

Ex.  5.  Given  the  bearings  and  distances  of  the  sides  of  a 
tract  of  land,  as  follow,— viz. :  1.  K  43°  25'  W.  28.43  ch. ; 

2.  K  29°  48'  E.  30.55  ch.;   3.  S.  80°  E.  28.74  ch.;  4.  K 
89°  55'  E.  40  ch. ;   5.  S.  10°  13'  E.  23.70  ch.;  6.  S.  63°  55' 
W.  25.18  ch.;   7. 1ST.  63° 45'  W.  20.82  ch.;   8.  S.  57°  25' W. 
31.70  ch. :  to  determine  the  area. 

Area,  262  A.,  2  E.,  31  P. 

Ex.  6.  Calculate  the  distances  of  the  third  and  fourth 
sides,  and  the  area  of  the  tract,  from  the  following  notes : — 
1.  S.  64°  5'  W.  11.18  ch. ;  2.  K  49°  45'  W.  12.91  ch. ;  3.  BT. 
35°  20'  E.,  distance  unknown;  4.  S.  82°  25'  E.,  distance 
unknown;  5.  ST.  87°  E.  13.82  ch. ;  6.  K  49°  30'  E.  4.95  ch. ; 
7.  S.  33°  25'  E.  10.80  ch. ;  8.  S.  0°  55'  E.  9.22  ch. ;  9.  S. 
79°  10'  W.  14.30  ch. ;  10.  K  52°  15'  W.  8.03  ch. 
Ans.  3d  side,  12.13  ch. ;  4th,  9.71  ch. ;  Area,  57  A.,  1  E,  12  P. 

Ex.  7.  One  corner  of  a  tract  of  land  being  in  a  swamp, 
but  visible  from  the  adjacent  corners,  I  took  the  bearings 
and  distances  as  follow:—!.  S.  45°  E.  13.65  ch.;  2.  K  38}° 
E.  17.28  ch. ;  3.  K  19°  W.  23.43  ch. ;  4.  S.  58°  W,  14  ch. ; 
5.  N.  87°  "W.  8.14  ch. ;  6.  K  45J-0  W.  9.23  ch. ;  7.  S.  28 J°  W. 
14.60  ch. ;  8.  S.  If  °  E. ;  9.  K  79 J°  E.  Eequired  the  distances 
of  the  last  two  sides  and  the  area  of  the  tract. 
Ans.  8th  side,  16.44  ch. ;  9th,  20.51  ch. ;  Area,  92  A.,  1  E.,  7  P. 

362,  Offsets.  If  any  of  the  sides  border  on  a  water- 
course, or  are  very  irregular,  stationary  lines  may  be  run  as 


236 


COMPASS   SURVEYING. 


[CHAP.  V. 


near  the  boundary  as  possible,  and  offsets  be  taken  aa 
directed  in  chain  surveying.  The  area  within  the  stationary 
lines  may  then  be  calculated  as  above.  That  of  the  spaces 
included  between  those  lines  and  the  true  boundary  is  to 
be  calculated  as  in  Art.  256.  These  areas  added  to  or 
subtracted  from  the  former,  according  as  the  stationary 
lines  are  within  or  without  the  tract,  will  give  the  content 
required. 

"When  the  tract  bounds  on  a  stream,  it  is  usual  to  con- 
sider the  boundary  as  the  middle  of  the  stream,  except  in 
tide  waters  or  large  rivers  which  are  navigable  and  are  thus 
considered  public  highways.  In  these  cases  the  boundary 
is  low-water  mark. 

In  reciting  the  boundaries  in  title-deeds,  the  offsets  are 
not  generally  given.  The  description  usually  runs  thus: 
—Thence  S.  43J°  E.  10.63  chains  to  a  stone  on  the  bank  of 
Ridley  Creek,  and  thence  on  the  same  course  1.05  chains 
to  the  middle  of  said  creek.  Thence  along  the  bed  of  said 
creek,  in  a  southwesterly  direction,  37.63  chains;  thence 
K".  47°  W.,  by  a  marked  white-oak  on  the  banks  of  the 
creek,  25.63  chains  to  a  limestone,  corner  of  John  Brown's 
land,  &c. 

EXAMPLES. 

Ex.  1.  Calculate  the  area  from  the  following  field-notes : — 


55 
55 

(4) 
1350 
0 

(3) 

t 

M£*fj& 

(6) 
1471 
930 

485 
0 

(5) 

S.SIOSO'E. 

55 
270 
396 
310 
340 
50 

(3) 
2160 
1929 
1408 
1015 
610 
0 
(2) 

60 
95 
140 
60 

N.5603(XE.      60 

(5) 
1072 
750 
390 
0 
(4) 

8.84045^. 

3050 
3000 

(1) 

loU 

Mid.  of  do.      85 
(2)011  r.bank.    55 
N.36°30'W. 

$6 

Middle 

75 

of  river. 

(7) 

S.45°15'W. 

75 

a 

100 

420 

60 

0 

(6) 

S.11°46'B. 

SEC.  IX.  J 


CONTENT  OF  LAND. 


237 


Sta. 

Bearings. 

Dist. 

N. 

S. 

E. 

W. 

E.  D.  D. 

W.D.D. 

Mult. 

N.Areas. 

S.  Areas. 

1 
2 

N.36>^W. 

30.00 

24.12 

li~.oT 

17.84 

47.96 

.OOE. 



N.56%E. 

21.60 

11.92 

.17 

.17E. 

2.0264 

3 

~T~ 

~5~ 

N.  26%  E. 

13.50 

12.06 

6.08 

24.09 

24.26E. 

292.5756 

S.  84%  E. 

10.72 

.98 

10.68 

16.76 

41.02E. 

40.1996 

S.51^E. 

14.71 

9.16 

11.51 

22.19 

63.21E. 

579.0036 

6 

S.  11%  E.  | 

8.26 

8.09 

1.68 

13.19 

76.40E. 

618.0760 

7 

s.  4514  w. 

42.41 

29.87 

30.12 

28.44 

47.96E. 

1432.5652 

48.10     48.10    47.96     47.96    76.40       76.40 


294.6020  2669.8444 


Area  of  offsets  calculated  as  in 
Ex.  1,  Art.  257. 


=       128.592265 


128  A.,  2  R.,  14.76  P. 

Ex.  2.  Given    the  field-notes    as   below   of   a   meadow 
bounding  on  a  small  brook,  to  calculate  the  area: — 


of 


(2) 

0 

1132 

55 

1054 

1       72 

896 

I       97 

739 

75 

480 

On  brook. 

^§T 

A 

63 

1414 

35 

1237 

87 

1016 

§  45 

824 

1  50 

652 

i 

551 

0 

452 

75 

295 

75 

0 

0 

(4) 
1396 


Ans.   34  A.,  3  K.,  0.6  P. 

Ex.  3.  Required  the  area  of  the  meadow  bordering  on  a 
mill-race,  of  which  the  boundaries  are  contained  in  the  fol- 
lowing field-notes,  the  angles  given  being  the  deflections 
from  the  last  course : — 


(2) 
11.28 

(D 


s.  53°i<yw. 


i 

OQ 

(3) 
21.65 

(2) 

r  97°  03' 

2.40 
1.96 
(3) 


to  race-bank. 

(4) 

r  97°  45'. 


238 


COMPASS   SURVEYING. 


[CHAP.  V. 


32 

9.89 

30 

5.50 

132 

3.00 

40 

1.08 

35 

0 

30°  12'  -| 

(5) 

(5) 

35 

1.05 

44 

.11 

(4) 

f81°14' 

(1) 

r-  98°  34' 

9.12 

(7) 

I-  27°  46' 

2.401 

corner  14 

2.26 

2.00 

0 

1.75 

6 

1,50 

32 

0 

12°  14^ 

(6) 

In  calculating  the  area,  it  will  be  necessary  first  to  calcu- 
late the  bearings  from  the  observed  angles. 

Area,  15  A.,  2  E.,  11.5  P. 

363.  Inaccessible  Areas.  When  it  is  desired  to  de- 
termine the  area  of  a  tract  of  difficult  access,  such  as  a 
pond,  a  thick  copse,  or  a  swamp,  it  should  be  surrounded 
by  a  system  of  lines  as  near  the  boundaries  as  they  can  be 
run  without  multiplying  the  number  of  sides  unnecessarily. 
Offsets  should  then  be  taken  to  different  points  of  the 
boundary,  so  as  to  determine  its  sinuosities.  The  areas  of 
the  parts  determined  by  these  offsets,  taken  from  the  area 
enclosed  in  the  base  lines,  will  .leave  the  content  required. 

"Where  two  base  lines  make 
an  angle  with  each  other,  the 
first  offset  on  each  should  be 
taken  to  the  same  point  in  the 
irregular  boundary.  Thus,  if 
AB  and  BC  (Fig.  162)  are  two 
adjacent  base  lines  enclosing 
an  irregular  boundary  HDI,  the 
first  offsets  should  be  taken  at  F  and  E,  so  situated  that  the 
offsets  FD  and  ED  should  meet  at  the  same  point  D  of  the 
boundary.  The  triangular  spaces  BDF  and  BDE  will  then 
be  included  with  the  areas  belonging  to  the  lines  AB  and 
BC  respectively. 


Fig.  152. 


SEC.  IX.] 


CONTENT  OF  LAND. 


239 


The  following  examples  of  the  field-notes  and  calculation 
for  the  area  of  a  pond  will  illustrate  this  subject: — 
Fig.  163  is  a  plat  of  Ex.  1  on  a  scale  of  1  inch  to  10  chains. 


Fig.  163. 


(3) 
1385 
1295 
1125 

975 
775 
430 
155 
47 
(2) 

155 

25 
0 
10 

55 

22 
75 
f78°55' 

(5) 
1866 
1805 
1675 
1475 
1250 
950 
800 
475 
0 

(4) 

115 

55 
90 
105 
22 
0 
42 
42 
T23°51' 

(1) 
1140 

1100 

875 
750 
500 
250 
75 
(6) 

r  f  52°  52'  on 
1  1(1).  (2). 

90 

10 
10 
60 
112 
112 
f  56°  35' 

(2) 
2280 
2215 
2015 
1770 
1480 
1240 
1000 
705 
555 
312 
55 

(1) 

60 
7 
33 
175 
291 
220 
32 
10 

(4) 
1152 
1135 

950 
775 
525 
250 
132 

(9 

42 
0 
0 
11 

70 
120 

T78°8' 

(6) 
920 
870 
750 
575 
300 
85 
(5) 

122 
32 
17 
73 

97 
f  69°  39' 

55 

82 
N.88°35'W. 

240 


COMPASS  SURVEYING, 


[CHAP.  V. 


Sta. 

Bearings. 

Dist. 

N. 

8. 

E. 

W. 

E.D.D. 

W.D.D. 

Multipli'r. 

S.  Areas. 

1 

N.88°35'W. 

22.80 

.55 

22.78 

29.88 

.00  E. 

2 

N.  9°  40'  W. 

13.85 

13.65 

2.33 

25.11 

25.11  W. 

342.7515 

N.68°28/E. 

11.52 

4.23 

10.72 

8.39 

16.72  W. 

70.7256 

4 

T~ 

~6~ 

S.  87°  41'  E. 

18.66 

.76 

18.64 

29.36 

12.64  E. 

9.6064 

S.  18°  2'  E. 

9.20 

8.75 

2.85 

21.49 

34.13  E. 

298.6375 

S.38°33'W. 

11.40 

8.92 

7.10 

4.25 

29.88  E. 

266.5296 

18.43  18.43 
Content  within  the  base-lines, 


32.21     32.21      59.24      59.24 


2)988.2506 
494.1253  ch. 


Base. 

Dist. 

Offsets. 

Inter. 
Dist. 

Sum  of 
Offsets. 

Areas. 

0.00 

0.55 

.82 

.55 

.82 

.4510 

3.12 

.55 

2.57 

1.37 

3.5209 

5.55 

.10 

2.43 

.65 

1.5795 

7.05 

.32 

1.50 

.42 

.6300 

10.00 

2.20 

2.95 

2.52 

7.4340 

12.40 

2.91 

2.40 

5.11 

12.2640 

(!)(2) 

14.80 

1.75 

2.40 

4.66 

11.1840 

1T.70 

.33 

2.90 

2.08 

6.0320 

20.15 

.07 

2.45 

.40 

.9800 

22.15 

.60 

2.00 

.67 

1.3400 

22.80 

0 

.65 

.60 

.3900 

45.8054 

0 

.47 

.75 

.47 

.75 

.3525 

1.55 

.22 

1.08 

.97 

1.0476 

4.30 

.55 

2.75 

.77 

2.1175 

(2)  (3) 

7.75 

.10 

3.45 

.65 

2.2425 

9.75 

0 

2.00 

.10 

.2000 

11.25 

.25 

1.50 

.25 

.3750 

12.95 

1.55 

1.70 

1.80 

3.0600 

13.85 

0 

.90 

1.55 

1.3950 

10.7901 

0 

1.32 

1.20 

1.32 

1.20 

1.5840 

2.50 

.70 

1.18 

1.90 

2.2420 

(3)  (4) 

5.25 

11 

2.75 

81 

2.2275 

7.75 

0 

2.50 

11 

.2750 

9.50 

0 

1.75 

0 

.0000 

11.35 

42 

1.85 

42 

.7770 

11.52 

0 

17 

42 

.0714 

7.1769 


SEC.  IX.] 


CONTENT  OF  LAND. 


241 


Base. 

Dist. 

Offset. 

Inter. 
Dist. 

Sum  of 
Offset. 

Areas. 

.00 

.42 

4.75 

.42 

4.75 

.84 

3.9900 

8.00 

.00 

3.25 

.42 

L3650 

9.50 

.22 

1.50 

.22 

.3300 

(4)  (5) 

12.50 

1.05 

3.00 

1.27 

3.8100 

14.T5 

.90 

2.25 

1.95 

4.3875 

16.T5 

.55 

2.00 

1.45 

2.9000 

18.05 

1.15 

1.30 

1.70 

2.2100 

18.66 

.00 

.61 

1.15 

.7015 

19.6940 

.00 

.85 

.97 

.85 

.97 

.8245 

3.00 

.73 

2.15 

1.70 

3.6550 

(5)  (6) 

5.75 

.17 

2.75 

.90 

2.4750 

7.50 

.32 

1.75 

.49 

.8575 

8.70 

1.22 

1.20 

1.54 

1.8480 

9.20 

.00 

.50 

1.22 

.6100 

10.2700 

.00 

.75 

1.12 

.75 

1.12 

.8400 

2.50 

1.12 

1.75 

2.24 

3.9200 

(«)(!) 

5.00 

.60 

2.50 

1.82 

4.5500 

7.50 

.10 

2.50 

.70 

1.7500 

8.75 

.10 

1.25 

.20 

.2500 

11.00 

.90 

2.25 

1.00 

2.2500 

11.40 

.00 

.40 

.90 

.3600 

Area  within  base  lines,  A.  49.41253 
Double  area,  cut  off  by 
(1)(2)  4.58054 

(2)  (3)      1.07901 

(3)  (4)      .H769 

(4)  (5)      1.96940 

(5)  (6)      1.02700 

(6)  (1)      1.39200 

J  of  10.76564  =  5.38282 
Area  of  pond,  44.02971  = 

16 


13.9200 


44 A.,  OR.,  4.75 P. 


242 


COMPASS*  SURVEYING. 


[CHAP.  V. 


The  following  are  the  field-notes  taken  for  the  survey  of 
a  pond.  The  area  is  required.  Fig.  164  is  the  plat,  to  a 
scale  of  1  inch  to  10  chains : — 


Fig.  164. 


Sta.  C 

f!460^ 

AB 

1090 

V  / 

JCm*U 

1050 

50 

Sta.E 

580 

937 

5 

1801 

560 

70 

675 
475 

275 

70 
85 
10 

1750 
1600 
1350 

50 

50  92°48n 

300 
150 
Sta.F 

15 

20 

150 

15 

1150 

50 

Sta.F 

45 
Sta.B 

95 

T  82°  16' 

900 
650 

10 

20 

627 
475 

0 
65 

Sta.B 

450 

5 

250 

0 

2100 

275 

25 

20 

90 

2035 
1890 

55 
5 

55 
Sta.D 

92  Q     ^onAB 

("627"). 

|-44°5' 

1660 
1500 

Sta.  A 
950 

r  f  87°!' 
I    \onAB. 

25 

60 

Sta.D 

842 

("1460) 

* 

900 

20 

>  __x 

Jx^^ 

785 

90 

750 

70 

^o27  ) 

<& 

500 

0 

585 

82 

650 

37 

400 

0 

350 

10 

400 

0 

300 

30 

185 

25 

225 

40 

150 

25 

20 

15 

15 

57 

27 

70 

Sta.  A 

S.66°37W. 

Sta.C 

f56°22' 

Sta.E 

|-70°29'. 

Area,  24  A.,  3  K.,  20  P. 


SEC.  IX.]  CONTENT  OF  LAND. 

364.   Compass  Surveying  by  Triangulation. 


243 


When  the  tract  is  bounded  by  straight  lines,  the  area 
may  be  found  by  determining  the  position  of  each  of 
the  angular  points  with  reference  to  one  or  more  base 
lines  properly  chosen. 

To  do  this,  measure  a  base  from  the  ends  of  which  all 
the  corners  of  the  tract  can  be  seen,  and  take  their  angles 
of  position.      There  will  thus  be   a   system  of  triangles 
formed,   giving    data    for    calcu- 
lating the   content  of  the  tract. 
Thus,  if  ABODE  (Fig.  165)  re- 
present a  field,  measure  a  base 
FG,  and  from  F  and  G  take  the    E 
bearings,  or  the  angles  of  posi- 
tion, of  A,  B,  C,  D,  and  E.     Cal- 
culate FA,  FB,  FC,  FD,  FE, 
and  thence  the  areas  of  the  tri- 
angles FAB,  FBC,  FCD,  FDE,       

and  FEA.  *  » 

Then,  ABODE  =  FBC  +  FCD  +  FDE  -  FEA  -  FAB. 

EXAMPLE. 

To  determine  the  area  of  a  field  ABODE,  I  mea- 
sured a  base  line  FG  of  12.25  chains,  and  at  F  and  G  I 
took  the  angles  of  position,  as  follow:— GFA  =  63°  15', 
GFB  =  27°  33',  GFC  =  35°  35',  GFD  =  58°  25',  GFE  - 
92°  10',  FGA  =  26°  5',  FGB  =  58°  30',  FGC  =  97°  12', 
FGD  =  72°  28',  and  FGE  =  37°  32'.  Fig.  165  is  a  plat 
of  this  tract,  on  a  scale  of  1  inch  to  10  chains. 


As  sin.  FAG 
:     sin.  FGA 
::   FG 
:     FA 


Calculation. 

t 

1.    To  find  FA, 
90°  40' 
26°    5' 
12.25 


.000029 
9.643135 
1.088136 
0.731300 


244  COMPASS  SURVEYING.  [CHAP.  V. 

To  find  FB. 

As  sin.  FBa  93°  57'  .001033 

:     sin.  BaF  58°  30'  9.930766 

:  :   Fa  1.088136 


:    FB  1.019935 

%  find  FC. 

As  sin.  FCa  47°  13'  0.134347 

:   sin.FaC  97°  12'  9.996562 

: :  Fa  1.088136 


:  FC  1.219045 

To  find  FD. 

As  sin.  FDa  49°    7'  0.121453 

:   sin.  FaD  72°  28'  9.979340 

: :  FG  1.088136 


:  FD  1.188929 

To  find  FE. 

As  sin.  FEa  50°  18'  0.113848 

:   sin.  FaE  37°  32'  9.784776 

::  Fa  1.088136 


FE  0.986760 

To  find  2  FAB. 

sin.  AFB  35°  42'  9.766072 

FA  0.731300 

FB  1.019935 


2  FAB  32.9084  1.517307 

To  find  2  FBC. 

sinBFC  8°    2'  9.145349 

BF  1.019935 

FC  1.219045 

2  FBC  24.2286  1.384329 


SEC.  IX.] 


CONTENT   OF  LAND. 


245 


sin.  CFD 
CF 
FD 
2  FCD 


sin.  DFE 
DF 
FE 
2  FDE 


sin.  AFE 
FE 
FA 
2FEA 


To  find  2  FCD. 

22°  50' 


99.2805 

To  find  2  FDE. 
33°  45' 


83.2585 

To  find  2  FEA. 
28°  55' 


25.2633 


2FBC 
2  FCD 
2  FDE 

2  FAB 
2FAE 


7  A.,  IE.,  28.76  P. 


32.9084 
25.2633 


9.588890 
1.219045 
1.188929 
1.996864 


9.744739 

1.188929 
0.986760 
1.920428 


9.684430 
0.986760 
0.731300 
1.402490 

24.2286 
99.2805 
83.2585 


206.7676 

58.1717 
2)148.5959 

74.29795  sq.ch. 


365,  If  no  two  points  can  be  found  from  which  all  the 
corners  can  be  seen,  several  points  may  be  taken,  and  these 
all  being  connected  by  a  system  of  triangles  with  a  single 
measured  base,  or  with  several  if  suitable  ground  for  mea- 
suring them  can  be  found,  the  area  may  then  be  calculated. 


246 


COMPASS  SURVEYING. 


[CHAP.  V. 


Thus,  (Fig.  166,)  if 
ABCDEFG  represent  a 
tract,  and  H,  I,  and  K, 
three  points  such  that, 
from  H,  B,  C,  D,  and  E, 
can  be  seen.  From  I,  all 
the  corners  can  be  seen, 
and  from  E!  we  can  see  A,  H 
G,  F,  and  E.  If  the  angles 
of  position  of  the  corners 
relatively  to  the  base  lines 
HI  and  HK  be  taken,  the 
distances  IA,  IB,  1C,  ID, 
&c.  may  be  found,  and  thence  the  areas  of  IAB,  IBC, 
ICD,  &c. 

Consequently,  ABCDEFG  =  ICD  +  IDE  +  IEF  + 
IFG  —  IGA  —  IAB  -  IBC  becomes  known. 

366.  The  same  principle  may  be  applied  to  surveying  a 
farm  by  means  of  one  or  more  base  lines  within  the  tract. 
If  such  lines  be  run,  and  the  corners  be  connected  by  triangles 
with  given  stations  in  these  bases,  the  tract  may  be  platted 
and  the  area  calculated. 

In  all  cases  of  the  application  of  this  method,  care  should 
be  taken  to  have  the  triangles  as  nearly  equilateral  as  possi- 
ble. If  any  of  the  angles  are  very  acute  or  very  obtuse, 
the  amount  of  error  from  any  mistake  in  measuring  the 
base  or  in  taking  the  angles  is  much  increased. 


CHAPTER  VI. 

TRIANGULAR    SURVEYING. 


367.  THE  method  pursued  in  the  last  few  articles  of' 
Chap.  Y.  constitutes  what  is  called  triangular  surveying.     It 
consists  in  connecting  prominent  points  with  one  or  more 
base  lines  by  means  of  a  system  of  triangles, — the  sides  of 
these  forming  bases  for  other  systems  until  the  whole  tract 
is  covered. 

The  positions  of  these  points  having  thus  been  accurately 
determined,  the  minuter  configurations  may  be  filled  up  by 
means  of  secondary  triangles,  or  by  any  of  the  other  methods 
of  surveying  of  which  we  have  already  treated. 

368,  Base.     In  triangular  surveying  there  is  generally 
but  a  single  base  measured  as  a  foundation  for  the  work. 
This  therefore  requires  to  be  measured  with  extreme  care, 
since  an  error  will  vitiate  the  whole  work.     The  precautions 
to  insure  extreme  accuracy  are  such  as  almost  to  preclude 
the  possibility  of  an  error.     Delambre,  in  speaking  of  a 
measurement  of  this  kind  in  France,  says, — 

"To  give  some  idea  of  the  precision  of  the  methods 
employed,  it  is  sufficient  to  relate  the  following  occurrence 
during  the  measurement  of  the  base  near  Perpignan:— One 
day,  a  violent  wind  seemed  every  moment  about  to  derange 
our  rules,  by  slipping  them  on  their  supports.  After  having 
struggled  a  long  time  against  these  difficulties,  we  finally 
abandoned  the  work.  Three  days  after,  on  a  calm  day,  we 
recommenced  the  work  of  that  whole  day,  and  we  only 
found  a  fourth  of  a  line  [one-twelfth  of  a  French  inch]  dif  • 

247 


248  TRIANGULAR  SURVEYING.  [CHAP.  VI. 

ference  between  two  measurements,  with  the  one  of  which' 
we  were  entirely  satisfied,  but  of  which  the  other  was 
esteemed  so  doubtful  that  we  considered  it  necessary  to 
perform  the  whole  work  anew." 

369.  Reduction  to  the  Level  of  the  Sea.    The  base 
should  if  possible  be  measured  on  level  ground.     A  smooth 
beach,  if  one  can  be  found  of  sufficient  length,  affords  one 
of  the  best  locations.     The  work  then  requires  no  further 
reduction.      If   the    ground  is   considerably  elevated,  the 
length  must  be  reduced  to  what  it  would  have  been  if  the 
same  arc  of  a  great  circle  had  been  measured  on  the  sea- 
'level.     This  will  be  shorter  than  the  measured  arc  in  the 
ratio  of  the  radius  of  the  circle  of  which  the  measured  arc 
forms  part  to  that  of  the  earth.     Thus,  suppose  the  arc  was 
on  ground  elevated  300  feet,  and  a  base  of  5000  yards  had 
been  measured:  then  say,  As  3956  miles  +  300  feet :  3956 
miles  : :  5000  yards  :  the  length  required. 

The  radius  used  should  be  that  belonging  to  the  latitude 
in  which  the  work  was  performed,  it  being  different  in  dif- 
ferent latitudes  in  consequence  of  the  oblateness  of  the  earth. 

370.  Signals.     The  base  having  been  measured,  the  next 
object  is  to  place  signals  on  prominent  points  over  the  coun- 
try.    Any  prominent  object  may  be  selected  for  this  pur- 
pose.    A  tree  on  a  hill,  provided  it  stands  so  that  its  trunk 
is  visible  projected  against  the  sky,  the  spire  of  a  church 
or  any  other  object  so  elevated  as  to  be  seen  from  a  great 
distance,  may  be  employed.     It  is  in  general  best,  however, 
to  employ  signals   constructed  expressly  for  the  purpose. 
Perhaps  one  of  the  best  is  a  tall  mast  with  a  flag  floating 
from  the  top.     The  flag  waving  in  the  wind  can  frequently 
be  seen  when  a  still  object  would  not  attract  the  attention. 
The  observation  must,  however,  be  taken  to  the  centre  of 
the  mast,  and  not  to  the  flag.    The  Drummond  light,  reflected 
in  the  proper  direction  by  a  parabolic  mirror,  is  the  best  of 
all.     Such  a  signal  may  be  seen  at  the  distance  of  sixty  miles. 

371.  Triangulation.     The  signals  having  been  placed, 


SEC.  IX.]  TRIANGULAR  SURVEYING.  249 

their  relative  position  should  then  be  determined  by  de- 
termining their  angles  of  position  with  each  other.  In  this 
triangulation  it  is  very  important  to  have  all  the  triangles 
as  nearly  equilateral  as  possible.  It  is  not  always  possible 
to  obtain  triangles  so  "well  conditioned"  as  would  be  de- 
sirable. The  rule  should,  however,  be  strictly  observed 
never  to  employ  a  triangle  with  a  very  acute  angle  opposite 
to  the  known  side,  as  a  very  small  error  in  4one  of  the 
adjacent  angles  will  then  produce  a  very  sensible  error  in 
the  calculated  distance.  For  example,  suppose  the  base 
AB  were  500  yards  long  and  the  adjacent  angles  were  A  = 
88°  39'  15"  and  B  =  88°  17'  45";  the  third  angle  C  would 
then  be  3°  3'. 

The  calculated  distance  of  AC  would  be  9394.6  yards : 
an  error  of  5"  in  one  of  the  observed  angles  would  cause 
an  error  in  this  result  of  half  a  yard, — a  quantity  utterly  in- 
admissible in  operations  of  this  nature. 

The  base  generally  being  short, 
compared  to  the  sides  of  the  tri- 
angles which  it  is  desirable  to 
employ,  these  should  be  gradually 
enlarged,  without  allowing  any  of 
them  to  become  "  ill  conditioned." 
The  mode  in  which  this  is  done 
may  be  seen  from  Fig.  167,  in 
which  AB  is  the  base,  on  which 
two  triangles  ABC  and  ABD,  both 
well  conditioned,  are  founded. 
The  line  CD  joining  their  vertices,  becomes  the  base  for 
two  other  triangles  DCE  and  DCF,  by  means  of  which  the 
line  EF  may  be  found. 

The  angles  at  all  the  points  of  the  triangle  should  be 
measured.  The  sum  of  these  should  be  180°.  If  it  differs 
but  little,  a  few  seconds,  from  this,  the  error  should  be  dis- 
tributed among  the  angles,  giving  one-third  to  each.  If, 
however,  a  greater  number  of  observations  have  been  made 
at  some  stations  than  at  others,  they  should  have  a  pro 
portionally  less  share  of  the  error.  Thus,  suppose  the 
recorded  angle  A  is  the  mean  of  5  observations,  B  the  mean 


250  TRIANGULAR  SURVEYING.  [CHAP.  VI. 

of  8,  and  C  of  2 :  $  =  £  of  the  error  should  be  applied  to 
A,  &  to  B,  and  ^  to  C. 

372.  Base  of  Verification.  In  order  to  prove  the  cor- 
rectness of  the  observations  and  calculations,  some  part  of 
the  work  as  distant  as  possible  from  the  base  should  be  con- 
nected with  another  carefully  measured  base, — the  coinci- 
dence of  the  measured  and  calculated  distance  of  which 
will  prove  the  whole  work.  In  a  system  of  triangulation 
carried  over  the  whole  of  France,  a  distance  of  more  than 
600  miles,  the  base  of  verification  was  found  to  be 

by  calculation  38406.54  feet  long, 

and     by  measurement  3840T.5 

The  difference  being  only  .96  feet, 

which  was  the  total  error  arising  from  observations  on  more 
than  sixty  triangles.  In  the  United  States  Coast  Survey, 
carried  on  under  the  supervision  of  Prof.  A.  D.  Bache,  still 
greater  accuracy  has  been  obtained. 


CHAPTER  VII.  - 

LAYING    OUT   AND    DIVIDING    LAND, 


SECTION  I. 

LAYING  OUT  LAND, 

Problem  1. — To  lay  out  a  given  area  in  the  form  of  a  square. 

373.  REDUCE  the  given  area  to  square  perches  or  square 
chains,  and  extract  the  square  root.     This  root  will  be  the 
length  of  one  side.     Erect  perpendiculars  at  the  ends  equal 
to  the  base,  and  the  thing  is  done. 

The  side  of  a  square  acre  is  316.23  links  =  12.65  poles 
=  69.57  yards. 

Problem  2. — To  lay  out  a  given  area  in  the  form  of  a  rect- 
angle, one  side  being  given. 

374.  Reduce  the  area  to  a  denomination  of  the  same 
name  as  the  side.     Divide  the  former  by  the  latter,  and  the 
quotient  will  be  the  length  of  the  other  side. 

EXAMPLES. 

Ex.  1.  Lay  out  10  acres  in  a  rectangular  form,  one  side 
being  12  chains  long.     Required  the  other  side. 

Ans.  8.33  chains. 

Ex.  2.  "What  must  be  the  length  of  one  side  of  a  rect- 
angle, the  area  being  15  acres  and  one  side  37.95  perches  ? 

Ans.  63.24  perches. 
251 


252  LAYING  OUT  AND   DIVIDING  LAND.  [CHAP.  VII. 

Problem  3. —  To  lay  out  a  given  area  in  a  rectangular  form, 
the  adjacent  sides  to  have  a  given  ratio. 

375.  Divide  the  given  area  expressed  in  square  chains  or 
square  perches  by  the  product  of  the  numbers  expressing 
the  ratio.  The  square  root  of  the  quotient  multiplied  by 
these  numbers  respectively  will  give  the  length  of  the  sides. 

DEMONSTRATION. — If  mx  and  nx  represent  the  sides,  and  A  the  area,  then 
will  mnx*  =  A.     Whence  x  =   I  — . 


EXAMPLES. 

Ex.  1.  Required  to  lay  out  an  acre  in  a  rectangular  form, 
so  that  the  length  shall  be  to  the  breadth  as  3  to  2.  What 
must  be  the  length  of  the  sides  ? 

Ans.  3.873  chains  and  2.582  chains. 

Ex.  2.  It  is  desired  to  lay  out  a  field  of  10  acres  in  a  rect- 
angular form,  so  that  the  sides  may  be  in  the  ratio  of  4  to  5. 
"What  are  these  sides  ? 

Ans.  8.944  chains  and  11.18  chains. 

Problem  4. — To  lay  out  a  given  area  in  a  rectangular  form, 
one  side  to  exceed  the  other  by  a  given  difference. 

376.  To  the  given  area  add  the  square  of  half  the  given 
difference  of  the  sides.  To  the  square  root  of  the  result 
add  said  half  difference  for  the  greater  side,  and  subtract  it 
for  the  less. 

CONSTRUCTION. — Make  AE  (Fig.  168)  equal  to  the 
given  difference  of  the  sides.  Erect  the  perpendicu- 
lar EG  equal  to  the  square  root  of  the  given  area. 
Bisect  AE  in  F,  and  make  FB  =  FG :  then  will  AB 
be  the  greater  side,  and  BE  the  less. 

For  (29.6)        AB  .  BE  =  EG*. 

The  rule  may  be  proved  thus :  FBa  &=  FG"  =  GEa 
-f-  EFa  =  area  -|-  square  of  half  the  difference  of 
the  sides.  Then,  AB  =  AF  -f  FB,  BC  =  FB  — 
FE. 


SEC.  I.]  LAYING  OUT  LAND.  253 

EXAMPLES. 

Ex.  1.  It  is  desired  to  lay  out  10  acres  in  the  form  of  a 
rectangle,  the  length  to  exceed  the  breadth  by  2  chains. 
Ans.  Length,  11.05  chains;  breadth,  9.05  chains. 

Ex.  2.  Required  to  lay  out  17  A.,  3  R.,  16  P.  in  a  rect- 
angular form,  so  that  one  side  may  exceed  the  other  by  50 
perches.  Ans.  Length  84,  and  breadth  34  perches. 

Problem  5. —  To  lay  out  a  given  area  in  the  form  of  a  tri- 
angle or  parallelogram,  the  base  being  given. 

377.  Divide  the  area  of  the  parallelogram,  or  twice  the 
area  of  the  triangle,  by  the  base.     At  any  point  of  the  base 
erect  a  perpendicular  equal  to  the  quotient.     The  summit 
will  be  the  vertex  of  the  triangle,  or  the  end  of  a  side  of 
the  parallelogram. 

If  through  the  summit  of  the  perpendicular  a  parallel  to 
the  base  be  drawn,  any  point  in  that  parallel  may  be  taken 
for  the  vertex  of  the  triangle. 

Problem  6. — To  lay  out  a  given  area  in  the  form  of  a  tri- 
angle or  parallelogram,  one  side  and  the  adjacent  angle  being 

given. 

378.  As  the  rectangle  of  a  given  side  and  sine  of  the 
given  angle  is  to  twice  the  area  of  the  triangle  or  the  area 
of  the  parallelogram,  so  is  radius  to  the  other  side  adjacent 
to  that  angle. 

DEMONSTRATION. — By  Art.  357  we  have  (Fig.  169) 
r  :  sin.  A  : :  AB  .  AC  :  2  ABC,  or  (1.6)  r  .  AB  :  sin.  A 
.  AB  : :  AB  .  AC  :  2  ABC ;  whence  sin.  A .  AB  :  2  ABC 
: :  r  .  AB  :  AB  .  AC  : :  r  :  AC.  % 


EXAMPLES. 

Ex.  1.  Required  to  lay  out  43  A.,  2  R.  in  the  form  of  a 
parallelogram,  one  side  AB  being  54  chains,  and  the  adja- 
cent angle  BAG  63°. 


254  LAYING  OUT  AND    DIVIDING  LAND.  [CHAP.  VII. 

/AB  54  A.  C.  8.267606 

As  Ai5-sin-  A\sin.  A  63°  «  0.050119 

:  ABCD  435  ch.  2.638489 

::  r  10.000000 

:       AC               9.04  ch.  1.956214 

Ex.  2.  Required  to  lay  out  3.5  acres  in  the  form  of  a  tri- 
angle, one  side  being  11.25  chains,  and  the  adjacent  angle 
73°  25'.  Ans.  AC  6.49  chains. 

Ex.  3.  Given  AB  K  85°  W.  16.37  chains,  BDS.  32 J°  W., 
to  determine  its  length  so  that  the  parallelogram  ABCD 
may  contain  16  acres.  Ans.  BD  =  10.99  chains. 

Ex.  4.  The  bearings  of  two  adjacent  sides  of  a  tract  of 
land  being  K  85°  E.  and  S.  23°  E.,  required  to  lay  off  10 
acres  by  a  line  running  from  a  point  in  one  side  14.37  chains 
from  the  angle  and  falling  on  the  other  side. 

Ans.  Distance,  14.63  chains. 


379.  Lemma.— If  ABC  (Fig.  170) 
be  any  triangle,  and  CD  a  line  in 
any  direction  from  the  vertex  cut- 
ting AB  in  D,  and  if  AF  be  taken 
a  mean  proportional  between  AB  A 
and  AD,  and  FE  be  drawn  parallel 
to  DC,  the  triangle  AFE  =  ABC. 


DEMONSTRATION. — Since  AD  :  AF  : :   AF  :   AB,   we  have 

(Cor.  2,  20.6)  AD  :  AB  : :  ADC  :  AFE ; 

but  (1.6)  AD  :  AB  : :  ADC  :  ABC, 
therefore  ABC  =  AFE. 

The  above  lemma  will  be  found  very  useful  in  the  con- 
structions required  in  dividing  land. 

The  mean  proportional  AF  may  be  found  by  describing 
a  semicircle  on  AD,  erecting  a  perpendicular  BG,  and 
making  AF  =  AG ;  or,  if  the  point  A  is,  remote,  we  may 
draw  BH  parallel  to  AC,  meeting  CD  in  H ;  draw  HI  per- 
pendicular to  CD,  cutting  the  semicircle  on  CD  in  I ;  make 


SEC.  L]  LAYING   OUT   LAND.  255 

CK  =  CI,  and  draw  KF  parallel  to  CA.  Then,  since  BH 
and  FK  are  parallel  to  AC,  the  line  AD  is  divided  similarly 
to  CD  (10.6) ;  but  CK  is  a  mean  proportional  between  CH 
and  CD,  therefore  AF  is  a  mean  proportional  between  AB 
and  AD. 

380.  Problem  7. — Two  adjacent  sides  of  a  tract  of  land 
being  given  in  direction,  to  lay  off  a  given  area  by  a  line  running 
a  given  course. 

Fig.  171. 

CONSTRUCTION. — Take  AD  (Fig.  171) 
any  convenient  length.     Erect  the  per- 

2  Area 

pendicular  AE  =  — A  _    .     Draw  the 
AD 

parallel  EF  cutting  AF  in  F.  Run  FG 
the  given  course.  Take  AB  a  mean  pro-  A  DBG 
portional  between  AD  and  AG  or  =  -v/AD  .  AG.  Then 
BC  parallel  to  GF  will  be  the  division  line. 

For,  by  construction,  ADF  =  the  given  area,  and,  by  lem- 
ma, ABC  =  ADF. 

AB  may  be  calculated  by  the  following  rule : — 

As  the  rectangle  of  the  sines  of  the  angles  adjacent  to 
the  required  side  is  to  the  rectangle  of  radius  and  the  sine 
of  the  angle  opposite  to  that  side,  so  is  twice  the  area  to  be 
cu^  off  to  the  square  of  that  side. 

The  truth  of  this  rule  is  evident  from  Art.  358. 

EXAMPLES. 

Ex.  1.  Given  AB  S.  63°  E.  and  AC  K  47°  15'  E.,  to  lay 
off  7  acres  by  a  line  BC  running  due  north.  Required  the 
distance  on  the  first  side. 


256 


LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 


Here  the  angles  are  A  =  69°  45',  B  =  63°,  and  C  =  47°  15'. 

Whence 


As 


{sin.  A 
sin.  B 
frad. 
1  sin.  C 
2  ABC 
AB2 
AB 


69°  45' 
63° 

47°  15' 
140  chains 

11.09 


Ar.  Co.    0.027709 

"     «      0.050119 

10.000000 

9.865887 

2.146128 

2)2.089843 

1.044921. 


Ex.  2.  Given  the  bearings  of  two  adjacent  sides,  taken  at 
the  same  station,  K".  57°  15'  "W.  and  1ST.  45°  30'  E.,  to  deter- 
mine the  distance  from  the  angular  point  of  a  station  on 
the  first  side  from  which  a  line  running  1ST.  77°  E.  will  cut 
off  5  acres.  Ans.  8.648  chains. 

Ex.  3.  Given  AB  S.  57°  E.  and  AC  S.  5°  16'  W.,  to  lay 
off  12  acres  by  a  line  running  N".  75°  E.  Required  the  dis- 
tance on  the  first  side.  Ans.  18.50  chains. 


381.  Problem  8.  —  The  directions  of  two  adjacent  sides  of  a 
tract  of  land  being  given,  to  lay  off  a  given  area  by  a  line  running 
through  a  given  point. 


Fis- 


CONSTRUCTION. — Divide  the  given 
area  by  the  perpendicular  distance 
from  P  to  AC,  (Fig.  172.)  Lay  off  ?- 
AD  equal  to  the  quotient.  Draw 
PE  parallel  to  AB.  Make  DF 
perpendicular  to  AD  and  equal  to 
AE.  Lay  off  FC  =  DE.  Then 
CPB  will  be  the  division  line. 


DEMONSTRATION.  —  Complete  the  parallelogram  ADHI. 

By  construction,  APD  is  half  the  required  area  ;  and,  therefore,  AIHD  con- 
tains the  required  area. 

Now,  because  the  triangles  PIB,  HPK,  and  CDK  are  similar,  and  their  homo- 
logous sides  IP,  DC,  and  HP  are  equal  to  the  three  sides  DF,  DC,  and  CF  of 
the  right-angled  triangle  DCF,  we  shall  have  (31.6)  HPK  =  PBI+  CDK.  To 


SEC.  I.]  LAYING  OUT  LAND.  257 

these  equals  add  AIPKD,  and  we  have  AIHD  =  ABC ;  whence  ABC  contains 
the  required  area. 

If  the  directions  of  AB  and  AC  and  the  position  of  the  point  P  be  given  by 
bearings,  AC  maybe  calculated  as  follows: — In  API  find  PI;  also  find  the 
perpendicular  PL.  Then  AD  =  area  -*-  PL.  Then  in  DFC  we  have  DF  =  PI 
and  FC  =  DE  to  find  DC,  which  added  to  AD  will  give  AC. 

If  FC  be  laid  off  on  both  sides,  another  point  C'  will  be  determined, 
through  which  the  line  may  run. 

EXAMPLES. 

Ex.  1.  Given  the  bearings  of  AB  K".  34°  W.,  and  of  AC 
West,  to  lay  off  18  acres  by  a  line  running  through  a  point 
P  bearing  from  A  K  41°  W.  18.85  chains. 


To  find  PL 

As  sin.  I 

56° 

A.  C.  0.081426 

:     sin.  PAI 

7° 

9.085894 

::  AP 

18.85 

1.275311 

:    PI 

2.77 

0.442631 

To  find  PL  and  AD. 
As  rad.  A.  C.  0.000000 

:    sin.  PAL  49°  9.877780 

:  :  PA  18.85  1.275311 

:    PL  1.153091 

Given  area,  180  ch.  2.255273 

AD  12.65  1.102182; 

whence  ED  =  AD  -  PI  =-  12.65  -  2.77  =  9.88. 

To  find  DC. 

FC  +  ED  =  12.65  1.102182 

EC  -  ED  =  7.11  0.851870 

2)1.954052 

DC  =  9.485  .977026; 

therefore  AC  =  AD  +  DC  =  12.65  +  9.485  =  22.135  ch. 

Ex.  2.   Given  the  angle  BAC  =  85°,  to  lay  off  6  acres 
by  a  line  through  a  spring  the  perpendicular  distances 

17 


258  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

of  which  from  AB  and  AC  are  3.25  chains  and  7.92  chains 
respectively.     Kequired  AC. 

Ans.   AC  ==  10.40  chains. 

Ex.  3.  A  has  sold  B  3J  acres,  to  be  laid  off  in  a  corner 
of  a  field,  by  a  line  through  a  tree  bearing  North  5.64 
chains  from  the  angular  point.  Now,  the  bearings  of  the 
sides  being  N.  46°  15'  E.  and  K  42°  W.,  it  is  required  to 
find  the  distance  to  the  division  line,  measured  on  the  first 
side.  Ans.  11.58  ch. 

382.  If  the  point  P  were  exterior  to  the  angle,  the  con- 
struction and  calculation  would  be  perfectly  analogous  to 
the  preceding.  The  following  is  an  example : — 

Given  the  angle  A  =  60°, 
(Fig.  173,)  EAP  =  90°,  and 
AP  =  23.42  chains,  to  cut 
off  14  A.  by  a  line  running 
through  P. 

Make  AD  =  ^|  =  5.98. 

Draw  PE  parallel  to  AB. 
Erect  the  perpendicular  DF 
=  AE,  and  make  FC  =  ED. 
Then  CB  will  be  the  divi- 
sion-line. 

For,  as  before,  AIHD  =  the  given  area;  but  PEH  = 
PBI  +  CKD ;  .-.  HIBK  =  CKD,  and  AIHD  =  ABC. 

r  :  tan.  30°: :  AP  (23.42)  :  AE  =  DF  =  13.52; 
whence  4        CF  =  DE  =  AE  +  AD  =  19.50, 
and  DC  =  ^/CF2-FD2  =  ^33.02x5.98  =  14.05 ; 

AC  =  5.98  +  14.05  =  20.03  chains. 

Problem  9.—  Three  adjacent  sides  of  a  tract  of  land  being 
given  in  position,  to  lay  off  a  given  area  in  a  quadrilateral  form 
by  i  line  running  from  the  first  side  to  the  third. 


SEC.  L] 


LAYING  OUT  LAND. 


299 


CASE  1. 

383.   The  division  line  to  be  parallel  to  the  second  side. 


Conceive  the  lines  CB  and 
DA  (Figs.  174,  175)  to  be  pro- 
duced till  they  meet,  and  cal- 
culate the  area  of  ABE.     Add 
this  to  the  given  area  if  the 
sum  of  the  angles  A  and  B  is 
greater  than  180°,  as  in  Fig. 
174 ;  but  if  the  sum  be  less, 
as    in    Fig.  175,   subtract  ] 
ABCD  from  ABE:  the  re- 
mainder will  be  the  area  of 
ECD. 


Fig.  174. 


Fig.  175. 


Then  say,  As  EAB  is  to 


ECD,  so  is  AB2  to  CD2. 
is  AB«vCDtoAD. 


And,  as  sin.  E  is  to  sine  of  B,  so 


The  following  is  a  neat  construction : — 

Produce  (0tB  and  GA  to  meet  in  E.  Erect  AF  perpen- 
dicular to  AB,  and  equal  to  double  the  area  divided  by  AB. 
Draw  FG  parallel  to  AB,  meeting  AE  in  G.  Then  the  tri- 
angle ABG  will  contain  the  required  area.  Take  ED  a 
mean  proportional  between  EA  and  EG,  or  let  ED  = 
>/EA.EG.  Through  D  draw  the  division  line  CD :  ABCD 
will  contain  the  required  area.  For  (lemma)  ECD  =  EBG ; 
whence  ABCD  =  ABG. 

The  calculation  is  more  concisely  made  by  the  following 
rule : — 

As  the  rectangle  of  the  sines  of  the  angles  A  and  B  is  to 
the  rectangle  of  radius  and  the  sine  of  E,  so  is  twice  the 
given  area  to  the  difference  between  AB2  and  CD2. 

This  difference,  added  to  AB2  when  the  sum  of  the 
angles  A  and  B  is  greater  than  180°,  but  subtracted  when 
the  sum  is  less,  will  give  CD2. 

Then,  As  sine  of  E  is  to  the  sine  of  B,  so  is  the  difference 
between  CD  and  AB  to  the  distance  AD. 


260  LAYING  OUT  AND  DIVIDING  LAND. 

DEMONSTRATION.  —     ECD  :     EBA  :  :  CDa   :    AB° ; 
Whence,  by  division,      ABCD  :     EBA  :  :  CD*  e*  AB*  :  AB" ; 
consequently, 
and 
But  (Art.  380) 


[CHAP.  VII. 


2  ABCD  :  2  EBA  :  :  CD3>»ABa  :  AB», 
2     EBA:      ABa::     2  ABCD     :CDafiN»AB". 
sin.  A.  sin.  B  :  rad.  sin.  E  : :     2  EBA  :  ABa; 


whence 


sin.  A.  sin.  B  :  rad.  sin.  E  : :  2  ABCD  :  CDa  *a  ABa. 


EXAMPLES. 

Ex.  1.  Given— 1.  K  62°  15'  E. ;  2.  1ST.  19°  12'  W.  7.92 
chains ;  3.  S.  87°  W.,  to  cut  off  5  acres  by  a  line  parallel 
to  the  second  side.  Required  the  length  of  the  division 
line,  and  the  distance  on  the  first  side. 

First  Method.— To  find  ABE,  (Art.358.) 


f  rad. 

A.  C.  0.000000 

I  sin.  E 

24°  45' 

"    "  0.378139 

(  sin.  A 

98°  33' 

9.995146 

\  sin.  B 

106°  12' 

9.982404 

JAB 

7.92 

0.898725 

1  AB 

0.898725 

2  ABE 

142.278 

2.153139 

2  ABCD 

100 

2  ECD 

242.278 

As  2  ABE 

142.278 

A.  C.  7.846861 

:    2  ECD 

242.278 

2.384314 

::  AB3 

J7.92 
17.92 

0.898725 
0.898725 

:    CD2 

2)2.028625 

CD 

10.335 

1.014312 

As   sin.  E 

24°  45' 

A.  C.  0.378139 

:    sin.  B 

106°  12' 

9.982404 

::  CD-AB 

2.415 

0.382917 

:    AD 

5.539 

0.743460 

Sic.  I.] 


LAYING  OUT  LAND. 


261 


As 


{sin. 
sin. 


Second  Method. 

98°  33' 
106°  12' 


100  ch. 

44.08T 
62.7264 


A.  C.  0.004854 
0.017596 
10.000000 
9.621861 
2.000000 
1.644311 


A 
B 

{rad. 
sin.  E 
:  :  2  ABCD 
:    CD2-AB2 

AB2  

Whence  CD  =  </ 106.8134  =  10.335,  as  before. 

Ex.  2.  Given— 1.  K  26°  47'  W. ;  2.  ST.  63°  13'  E.  12.72 
chains ;  3.  S.  8°  17'  E.,  to  cut  off  7  acres  by  a  line  parallel 
to  the  second  side.  The  distance  on  the  first  side  and  the 
length  of  the  division  line  are  required. 

Ans.   Division  line,  10.72  chains;  distance,  5.98  ch. 

Ex.  3.  Given  the  bearing  of  three  sides  of  a  tract  of 
land,  and  the  length  of  the  middle  one,  as  follow, — viz. :  1. 
ET.  15°  30'  W. ;  2.  K  74°  30'  E.  11.60  chains ;  3.  S.  45°  E. : 
to  cut  off  12  acres  by  a  line  parallel  to  the  second  side. 
The  division  line  and  distance  on  the  first  side  are  re- 
quired. 

Ans.  Division  line,  16.44  chains;  distance,  8.555  ch. 

384.  If  AD  and  BC  (Fig.  176)  are  «&  w. 

nearly  parallel,  the  following  method  may 
be  employed  with  advantage : — 

Divide  the  area  by  AB :  the  quotient 
will  give  the  approximate  length  of  the 
perpendicular  AI.  Draw  FE  parallel  to 
AB,  and  AK  parallel  to  BH.  In  AIK 
and  AIF  find  IK  and  IF. 

FK  =  FI  ±  IK,  and  FE  =  AB  ±  FK. 
If  the  sum  of  the  angles  is  greater  than  180°,  the  area  cut 
off  by  EF  will  be  too  great  by  the  small  triangle  AFK  = 
FK.AI  AFK      FK.AI 


Make  IL 


Then  will  AL  be 


2  FE          2  FE 

the  corrected  perpendicular :  AD  may  thence  be  found. 


262  LAYING  OUT  AND  DIVIDING  LAND.          [CHAP.  VII. 

EXAMPLES. 

Ex.  1.  Given  GA  K  87°  W.,  AB  K".  5°  W.  14.25  chains, 
and  BH  S.  89°  E.,  to  lay  off  10  acres  by  a  line  parallel  to 
AB. 

Here  the  angles  are  A  =  98°  and  B  =  84° :  AK  will 
therefore  lie  between  I  and  F. 

AI  =  -T-TT^  ~  f -02  chains,  nearly. 
14.,2o 

In  IAF  we  have  IAF  =  8°  and  IA  =  7.02 ;  whence  IF  = 
.987. 

In  IAK  we  have  IAK  =  6°  and  IA  =  7.02 ;  whence  IK  = 
.738. 


Whence 
Hence 

and 
whence 

KF 
IL 

AL 
AD 

= 

7 
7 

25  and  EF  = 
KF  .  AI 

14 
'6< 

96 

.50. 
chains  ; 

2EF 

.02  -.06  =  6. 
.03  chains. 

The  above  method  is  very  convenient  for  field  operations. 
EF  may  be  measured  directly  on  the  ground;  whence  FK  is 

FIT      AT 

known,  and  IL  =  — ^TFT~>  as  before. 

Ex.  2.  Given  GA  North,  AB  1ST.  89°  E.  7.86  chains,  and 
BO  8.  1°  30'  W.,  to  cut  off  10  acres  by  a  line  parallel  to 
AB.  Eequired  the  distance  of  the  division  line  from  A. 

Ans.   13.00  ch. 

CASE  2. 

385.  By  a  line  running  a  given  course. 


Construct,  as  in  last  case, 
ABG  to  contain  the  given 
area.  Draw  BL  according 
to  the  given  course.  Take 
ED  a  mean  proportional  B"  AWL 


SEC.  I.]  LAYING   OUT   LAND.  263 

between  EL  and  EG :    CD  p  Fig.  us. 

parallel  to  BL  will  be  the 

division  line.     For,  by  the 

lemma,      ECD    =    EBG; 

whence    ABCD  =  ABG, 

the  required  area. 


A*^  W        UT     \D    G 
*"^J          *» 

:••:;  *'"A 

The  calculation  may  be  performed  by  the  finding  .ATC  and 
the  area  of  ABE ;  whence  ECD  becomes  known.  The  dis- 
tance ED  may  then  be  found  by  Art.  380 ;  or, 

Conceive  Wn  to  be  drawn  parallel  to  CD,  making  EWn 
=  EAB.  Then  say,  As  the  rectangle  of  the  sines  of  the 
angles  C  and  D  is  to  the  rectangle  of  the  sines  of  A  and  B, 
so  is  the  square  of  AB  to  the  square  of  Wn. 

And,  As  the  rectangle  of  the  sines  of  C  and  D  is  to  the 
rectangle  of  radius  and  sine  of  E,  so  is  twice  the  given  area 
to  a  fourth  term. 

If  the  sum  of  the  angles  A  and  B  is  greater  than  180°, 
add  these  fourth  terms  together ;  but,  if  the  sum  of  A  and 
B  is  less  than  180°,  subtract  the  second  fourth  term  from  the 
first :  the  result  will  be  the  square  of  the  division  line  CD. 

Then,  As  sine  of  C  is  to  sine  of  B,  so  is  AB  to  a  fourth 
term ;  take  the  difference  between  this  fourth  term  and  CD, 
and  say,  As  sine  of  E  is  to  the  sine  of  C,  so  is  this  dif- 
ference to  AD. 

DEMONSTRATION. — Since  EnW  =  EAB,  EW  is  a  mean  proportional  between 
E  A  and  EL.     Whence  riW  is  a  mean  proportional  between  AP  and  BL ;  there- 
fore AP  .  BL  =  nWa. 
Now,  by  similar  triangles,  we  have 

sin.  L  (sin.  D)  :  sin.  A  : :  AB  :  BL, 

and  sin.  P  (sin.  C)  :  sin.  B  : :  AB  :  AP. 

Whence  (23.6)  sin.  C  .  sin.  D  :  sin.  A  .  sin.  B  : :  AB9  :  AP  .  BL  =  nW»; 
and,  by  demonstration  to  last  case, 

sin.  C  .  sin.  D  :  rad.  sin.  E  : :  2  wWDC  :  CDac«nWa. 
Draw  AMN  parallel  to  BC.     Then,  in  the  triangle  ABM,  we  have 

sin.  M  (sin.  C)  :  sin.  BAM  (sin.  B)  : :  AB  :  BM ; 
and,  in  AND,  we  have 

sin.  NAD  (sin.  E)  :  sin.  N  (sin.  C)  : :  DN  (CD  *cBM)  :  AD. 


264  LAYING  OUT  AND  DIVIDING  LAND.          [CHAP.  VII. 

EXAMPLES. 

Ex.  1.  Given— 1.  ST.  62°  15'  E. ;  2.  N.  19°  12'  W.  7.92 
chains ;  3.  S.  87°  W.,  to  cut  off  5  acres  by  a  line  perpen- 
dicular to  the  first  side.  Required  the  length  of  the  divi- 
sion line,  and  its  distance  from  the  end  of  the  first  side. 


First  Method. 

As  sin.  E 

24°  45' 

Ar.  Co.  0.378139 

:   sin.  B 

106°  12' 

9.9S2404 

::  AB 

7.92 

0.898725 

:   EA 

18.166 

1.259268 

AB 

0.898725 

sin.  A 

98°  33' 

9.995146 

2  ABE 

142.278 

2.153139 

2ABCD 

100 

2ECD 

242.278 

Then,  (Art.  380,) 

r  sin.  E 

24°  45' 

Ar.  Co.    0.378139 

As1    -     -n 
(  sin.  D 

90° 

"    "     0.000000 

(  rad. 

10.000000 

'    I  sin.  0 

65°  15' 

9.958154 

:  :  2  ECD 

242.278 

2.384314 

:  ED2 

2)2.720607 

ED 

22.93 

1.360303 

AE 

18.17 

AD 

4.76 

As  sin.  C 

65°  15' 

Ar.  Co.  0.041846 

:    sin.  E 

24°  45' 

9.621861 

::  ED 

1.360303 

:    CD 

10.57 

1.024010 

SEC.  L]  LAYING  OUT  LAND.  265 


Second  Method. 

f  sin.  C 

65°  15' 

Ar.  Co.  0.041846 

(  sin.  D 

90° 

"     «    0.000000 

r  sin.  A 
{  sin.  B 

98°  33' 
106°  12' 

9.995146 
9.982404 

rAB 

7.92  chains 

0.898725 

:       nW2 

65.5913 

0.898725 

1.816846 

f  sin.  C 

Ar.  Co.    0.041846 

\  sin.  D 

«     «      0.000000 

f  rad. 
\  sin.  E 

24°  45' 

10.000000 
9.621861 

::  2ABCD 

100  chains 
46.1006 

2.000000 

1.663707 

CD!    - 

65.5913 

=  10.57. 

-/111.6919  = 

As  sin.  C 

65°  15' 

Ar.  Co.  0.041846 

:    sin.  B 

106°  12' 

9.982404 

::  AB 
:   BM 

7.92 
8.375 

0.898725 

0.922975 

CD 

10.57 

DN       ^ 

2.195 

As  sin.  E 

24°  45' 

Ar.  Co.  0.378139 

:    sin.  C 

65°  15' 

9.958154 

:    AD 

2.195 
4.76 

0.341435 

0.677728 

266  LAYING  OUT   AND  DIVIDING  LAND.  [CHAP.  VII. 

It  will  be  seen  from  the  above  that  the  first  method  is  in 
this  case  the  shorter.  It  has  the  advantage,  also,  of  first 
giving  the  value  of  AD,  which  of  itself  is  sufficient  to  de- 
termine the  position  of  the  division  line. 

In  the  second  method,  if  AG  and  BH  are  nearly  parallel, 
the  calculation  for  CD  and  DN  should  be  carried  to  the 
third  decimal  figure. 

The  construction  given  for  this  and  the  preceding  case 
admits  of  easy  application  on  the  ground. 

Run  the  lines  CB  and  GA  to  their  point  of  intersection ; 
lay  out  the  perpendicular  AF ;  run  FG  parallel  to  AB  and 
BL  parallel  to  the  division  line.  Measure  EL  and  EG,  and 
make  ED  =  v/EL  .  EG. 

Ex.  2.  The  bearings  of  three  adjacent  sides  of  a  tract  of 
land  are— 1.  K  26°  47'  W. ;  2.  K  63°  13'  E.  12.72  chains ; 
3.  S.  8°  17'  E.,  to  cut  off  7  acres  by  a  line  running  due 
east.  The  distance  on  the  first  side  and  the  length  of  the 
division  line  are  required. 

Ans.  Distance,  3.37 ;  division  line,  11.11. 

Ex.  3.  The  bearings  of  three  adjacent  sides  of  a  tract  of 
land  being— 1.  K  78°  17'  E;  2.  N.  5°  13'  E.  15.62  chains; 
and  3.  K  63°  43'  W.,  it  is  desired  to  cut  off  10  acres  by  a 
line  making  equal  angles  with  the  first  and  third  sides. 
What  is  the  bearing  of  the  division  line,  and  its  distance 
from  the  end  of  the  first  side  ? 

Ans.  Bearing,  K  7°  17'  E. ;  distance  on  first  side,  6.316. 

If  the  first  and  third  sides  are  nearly  parallel,  the  area  of 
ABL  may  be  calculated.  This  taken  from  ABCD,  or 
added  to  it,  according  as  BL  falls  within  or  without  the 
tract,  will  give  the  area  of  BLDC,  which  may  be  parted  off 
as  directed  in  Art.  384. 


SEC.  1.] 


LAYING  OUT   LAND. 


267 


CASE  3. 

386.  By  a  line  through  a  given  point. 

Produce    CB    and   DA  Fig.  179. 

(Fig.  179)  to  meet  in  E, 
and  calculate  the  area 
EAB.  Thence  ECD  is 
found.  Proceed  as  in  Art. 
381.  Thus,  calculate  or 
measure  the  perpendicular 

FOT) 

PL     Lay  off  EF  =  -p— . 

Draw  PK  parallel  to  BE, 
meeting  AE  in  K.  Erect  the  perpendicular  FG  =  EK  or 
BP,  and  make  GD  =  FK.  Then  will  the  division  line  pass 
through  D. 

Calculation. 

Determine  AE.  Then  ED  =  EF  +  </FK2  —  EK2,  and 
AD  =  ED  -  EA. 

EXAMPLES. 

Ex.  1.  Given  DA  West,  AB  K  16°  15'  W.  6.30  chains, 
BC  K  57°  E.,  to  cut  off  3  acres  by  a  line  through  a 
spring  P,  situated  K  25°  30'  E.  6.09  chains  from  the 
corner  A. 

To  find  EA,  EAB,  and  ECD. 

Ar.  Co.  0.263891 
9.981171 
0.799341 
1.044403 
0.799341 
9.982294 
1.826038 


As 

sin.  E 

33° 

: 

sin.  B 

73°  15' 

:  : 

AB 

6.30 

: 

EA 

11.077 

AB 

6.30 

sin.  A 

73°  45' 

2  EAB 

66.994 

2  ABCD 

60. 

2  ECD  = 

126.994. 

268  LAYING  OUT  AND   DIVIDING  LAND.  [CHAP.  VII. 

To  find  PI  and  EF. 

As  rad.  Ar.  Co.  0.000000 

:    sin.PAI  64°  30'  9.955488 

::  AP  6.09  0.784617 

:   PI  5.497  0.740105 

ECD  63.497  1.802753 

EF  11.552  1.062648 

To  find  AK,  EK,  and  KF. 

As  sin.  K  33°  Ar.  Co.  0.263891 

:    sin.  APK  31°  30'  9.718085 

::  AP                        6.09  0.784617 

:    AK                        5.842  0.766593 

AE  11.077 
EK  =  FG  =         5.235 
Whence         KF  =  GD  =  EF  -  EK  =  6.317. 

To  find  FD. 

GD  +  GF  11.552  1.062648 

GD  -  GF  1.082  0.034227 

2)1.096875 

FD  =  3.535  .548437 

Whence         AD  =  EF  +  FD  —  EA  =  4.01. 

Ex.  2.  The  bearings  of  three  adjacent  sides  of  a  tract  of 
land  are  as  follow,— viz. :  DA  1ST.  47°  E.,  AB  K  35°  16'  W. 
15.23  chains,  and  BC  8.  36°  W.,  to  cut  off  15  acres  by  a 
line  running  through  a  spring  P  9.22  chains  distant  from 
the  first,  and  10.55  chains  from  the  second,  side.  The  dis- 
tance of  the  division  line  from  the  end  of  the  first  side  is 
required.  Ans.  10.82  chains  from  A. 


SEC.  L] 


LAYING  OUT  LAND. 
CASE  4. 


269 


387.  By  the  shortest  line. 

Produce  the  lines  CB  and  DA 
(Fig.  180)  to  meet  in  E,  and  calcu- 
late ABE  and  AE,  whence  ECD  is 
known.  ~Now,  the  shortest  line  cut- 
ting off  a  given  area  will  make  equal 
angles  with  the  sides.  Therefore  EG 


Fig.  180. 


ED. 

ED2,  sin  E 
R 


R 


whence  we  must  have  AD  =  EA 


R.2ECD 


sin.E 

Or,  this  case  may  be  constructed  and  calculated  as  Case  2 
by  drawing  BL  so  as  to  make  the  angles  EBL  andELB  equal. 

Ex.  1.  Given  DA  K  86°  W.,  AB  K  19°  20'  E.  16.75  ch., 
and  BC  K  63°  30'  E.,  to  cut  off  15  acres  by  the  shortest 
line.  The  distance  on  AD  and  the  bearing  of  the  division 
line  are  required. 

AD  =  13.38;  bearing  of  DC,  K  11  J°  W. 

Problem  10. — To  cut  off  a  plat  containing  a  given  area  from 
a  larger  tract  of  any  number  of  sides. 

CASE  i. 

388.  When  the  division  line  is  to  be  drawn  from  one  of  the 
angles. 


Find  by  trial  the  side  EF  (Fig. 
181)  on  which  the  division  line  will 
fall,  and  calculate  the  area  ABCDE : 
subtract  this  area  from  that  re- 
quired; the  remainder  will  be  the  c 
area  of  AEG,  which  may  be  laid  off 
as  in  Prob  6,  Art.  378.  Or, 

The  course  and  distance  may  be 
calculated  directly  as  follows : — 


Fig.  181. 


270 


LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 


Change  the  bearings  so  that  the  side  on  which  the  division 
line  will  fall  may  be  a  meridian. 

Take  out  the  latitudes  and  departures.  The  difference 
between  the  sums  of  the  eastings  and  westings  will  be  the 
departure  of  the  division  line. 

Find  the  multipliers,  assuming  that  corresponding  to  the 
division  line  to  be  0. 

Multiply  the  known  latitudes  by  the  multipliers,  and 
place  the  products  in  the  columns  of  areas. 

Subtract  the  difference  of  the  sums  of  the  ndrth  and  south 
areas  from  double  the  required  area :  the  remainder  will  be 
the  area  corresponding  to  the  side  on  which  the  division 
line  will  fall.  Divide  this  area  by  the  multiplier:  the 
quotient  will  be  the  latitude  of  that  side.  Place  it  in  its 
proper  column. 

Take  the  difference  between  the  sums  of  the  northings 
and  southings :  this  difference  will  be  the  latitude  of  the 
division  line.  "With  this  latitude  and  the  departure  before 
determined  calculate  the  distance  and  changed  bearing, 
from  which  the  real  bearing  is  readily  determined. 

EXAMPLE. 

Ex.  1.  Let  the  bearings  and  distances  be  as  follows: — 
1.  S.  47}°  W.  12.21  ch. ;  2.  K  49°  W.  15.28  ch. ;  3.  K  13°  E. 
13.18  ch. ;  4.  S.  76J°  E.  17.95  ch. ;  5.  S.  89f  °  E.,  to  cut  off 
35  acres  by  a  line  from  the  first  angle  and  falling  on  the  last 
side.  Required  the  distance  on  the  last  side. 

First  Method. 


AB 
BC 

Bearings. 

Dist. 

N. 

S. 

E. 

W. 

E.  D.D. 

W.D.D. 

Mult. 

N.Areas 

S.  Areas. 
205.7106 

S.47^°W. 

12.21 

8.25 

9.00 

8.88 

0000 

N.49°W. 

15.28 

10.02 

11.53 

20.53 

20.53  W. 

CD 
Dl 
EA 

N.13°E. 

13.18 

12.84 

2.96 

8.57 

29.10  W. 

373.6440 

S.76%°E. 

17.95 

4.19 

17.45 

20.41 

8.69  W. 

36.4111 

(10.42) 

(     .12) 

17.57 

8.88  E. 

92.5296 

22.86     22.86     20.53    20.53    37.98      37.98                                    671.8842 
36.4111 

2  ABODE      635.4731 

2ABCDEG    700 

2  AEG  64.5269 


SEC.  L] 

As  diff.  lat.  EA 

:    dep. 

::   rad. 

:   tan.  bear.  EA 
Bear.  EF 
AEF  = 

As  cos.  bearing 

:    rad. 
::    diff.  lat. 

:   dist. 

Then,  (Art.  378,) 
A     f  AE 

1  t  sin.  AEG 
:        2  AEG 
: :         r 
:       EG 


LAYING  OUT  LAND. 


271 


10.42 
.12 

S.  0°40/E. 

S.  89°  45'  E. 

89°  5r 

0°40' 


10.42 


A.  C.  8.982132 

1.079181 

10.000000 

8.061313 


A.  C.  0.000029 

10.000000 

1.017868 

1.017897 


10.42 

A.  C.  8.982103 

89°  5' 

«  «  0.000056 

64.5269 

1.809741 

10.000000 

6.19 


0.791900 


272  LAYING  OUT  AND  DIVIDING  LAND.          [CHAP.  VII. 


w 


00 


i£   O3 

£S 


S 


r-t       00 

cq     o 


SH. 

(M 


M 


M 


0     ft 


SEC.  I.] 


LAYING  OUT  LAND. 


273 


Ex.  2.  Given  as  follows:—!.  K  27  J°  W.  5  ch.;  2.  K  58° 
W.  9.53  ch. ;  3.  K  42}°  E.  9.60  ch.;  4.  S.  81J°  E.  14  ch.; 
5.  S.  28  J°  E.:  to  lay  off  25  acres  by  a  line  from  the  first 
station.  The  distance  on  the  fifth  side  is  required. 

Ans.  10.76  ch. 

CASE  2. 
389.  The  division  line  to  run  a  given  course. 

Proceed  as  in  Case  1  to  find  the  area  of  the  tract  to  a  line 
through  the  ends  of  the  sides  on  which  the  division  line 
will  fall,  and  the  bearing  and  distance  of  the  closing  line. 
The  difference  between  this  area  and  the  area  to  be  laid  off 
will  be  the  area  of  a  quadrilateral  which  may  be  parted  off 
as  in  Art.  385. 


EXAMPLES. 

Ex.  1.  The  boundaries  of  a  tract  of  land  are  as  follows, — 
viz. :  1.  K  75°  E.  13.70  ch. ;  2.  K  20J°  E.  10.30  ch. ;  3.  East 
16.20  ch. ;  4.  S.  33£°  W.  35.20  ch. ;  5.  S.  76°  W.  16.00  ch. ; 
6.  North  9.00  ch. ;  7.  S.  84°  W.  11.60  ch. ;  8.  K  53J°  W. 
11.60  ch. ;  9.  K  36f  °  E.  19.60  ch. ;  10.  K  22}°  E.  14.00  ch. ; 
11.  S.  76  j°  E.  12.00  ch. ;  12.  S.  15°  W.  10.85  ch. ;  13.  S.  18° 
W.  10.62  ch.  It  is  required  to  lay  off  35  acres  from  the 
eastern  end  of  the  farm  by  a  line  perpendicular  to  the  first 
side.  The  distance  of  the  division  line  from  the  second 
corner  is  required. 

Fig.  182. 


Fig.  182  is  a  plat  of 
this  tract. 


274  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

To  find  BODE  and  the  bearing  and  distance  of  EB. 


Sta. 
~BC 

Bearings. 

Dist. 

N. 

S. 

E. 

W. 

E.D.D. 

W.D.D 

Multipl'r. 

Areas. 

N.  20^°  E. 

10.30 

9.65 

3.61 

3.23 

.00  E. 

CD 

East. 

16.20 

16.20 

19.81 

19.81  E. 

DE 
EB 

S.  33%°  W. 

35.20 

29.35 

19.43 

3.23 

16.58  E. 

486.6230 

19.70 

.38 

19.81 

3.23  W. 

63.6310. 

29.35    29.35     19.81    19.81      23.04      23.04                      2)550.2540 
275.1270 

Latitude  of  EB  19.70 

Departure  of  EB  .38 

Tangent  of  bearing  K  1°  6'  W. 


Cosine  of  bearing 
Latitude 
Distance  EB 


19.70 


A.  C.  8.705534 

-  1.579784 
8.285318 

A.  C.  0.000080 
1.294466 
1.294546 


Now,  AB  differing  in  course  from  FE  by  only  1°,  the  fol- 
lowing is  the  best  method  of  determining  the  position  of 
the  division  line  OP,  which,  by  the  conditions,  is  to  be  per- 
pendicular to  AB. 

Draw  ET  perpendicular  to  AB,  and  find  ET  and  BT  : 


then  will  BO  =  J  BT  + 


ET 


very  nearly. 


cos.  EBT 
EB 
BT 


To  find  BT  and  EF. 
76°  6' 


sin.  EBT 
EB 
ET 
OBEP  =  350  -275.1270 


OB 


4.733 


19.127 

74.873 

3.915 

2.366 

6.281 


9.380624 
1.294546 
0.675170 

9.987092 
1.294546 
1.281638 
1.874325 
0.592687 


SEC.  L]  LAYING  OUT  LAND.  275 

Ex.  2.  The  boundaries  of  a  tract  of  land  being  as  follow, — 
viz.:  1.  K  39°  E.  12.17 chains;  2.  S.  88J °  E.  14.83  chains; 
3.  K  67J°  E.  13.32  chains;  4.  S.  27J°  E.  J6.67  chains ;  5.  6. 
57J°  W.  21.92  chains ;  6.  S.  73°  W.  18.23  chains ;  7.  S.  52J° 
"W.  12.00  chains;  8.  K  37°  W.  22.72  chains;  9.  K  67|° 
E.  18.00  chains, — to  cut  off  55  acres  from  the  east  end  by«i 
line  bearing  S.  37°  E.  Required  the  position  of  the  point 
at  which  the  line  must  commence. 

Ans.    On  the  first  side,  at  9.21  chains   from  the  be- 
ginning. 

Problem  11. — To  straighten  boundary  lines. 

390.  It  often  becomes  necessary  to  straighten  crooked 
boundaries  between  farms,  so  as  to  leave  the  same  quantity 
of  land  in  each  farm. 

First  Method.— If  the  tracts  Fig.  183. 

^"i 

are  platted,  this  may  be  done 
approximately  by  parallels. 
Thus,  suppose  BODE  (Fig. 
183)  was  the  common  bound- 
ary of  two  farms,  and  it  is 
agreed  by  the  owners  to  run 
a  straight  fence  from  B  to 
fall  somewhere  on  EG.  Join 
CE,  and  draw  DK  parallel  to 
it;  then  join  BK,  and  draw  CL  parallel  thereto:  BL  will 
be  the  line  required.  In  open  ground,  this  work  may  be 
performed  in  the  field  by  the  transit. 

391.  Second  Method. — Where  the  lines  are  straight,  the 
method  of  latitudes  and  departures  will  enable  us  to  run 
the  line  with  accuracy.     For  it  is  evident  that,  if  we  cal- 
culate the  area  contained  by  the  boundaries  BCDELB,  it 
should  be  0,  since  the  new  line  is  intended  to  add  to  the 
contents  of  neither  farm.     The  calculation  would  therefore 
be  precisely  the  same  in  principle  as  in  Art. 


276  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

EXAMPLES. 

Ex.  1.  Given  BC  S.  61°  E.  16.50  chains ;  CD  K  53J°  E. 
20.05  chains ;  DE  S.  51°  E.  18.42  chains ;  EG  K  10J°  E. 

Eule  a  table  as  below.  Then  change  the  bearing  so  that 
the  side  on  which  the  new  line  will  fall  shall  be  a  meridian. 
Take  out  the  latitudes  and  departures :  the  difference  be- 
tween the  sums  of  the  eastings  and  westings  will  be  the 
departure  of  the  new  line.  Find  the  double  departures 
and  the  multipliers,  assuming  that  corresponding  to  the 
first  side  equal  to  its  double  departure :  that  corresponding 
to  the  division  line  will  thus  be  0.  Find  the  areas :  the 
difference  between  the  north  and  the  south  areas  will  be 
the  area  corresponding  to  the  side  on  which  the  line  will 
fall.  Divide  this  area  by  the  multiplier  of  that  side :  the 
quotient  will  be  the  difference  of  latitude  of  that  side, 
which,  as  the  changed  bearing  was  north,  will  also  be  equal 
to  its  distance.  By  balancing  the  latitudes  we  may  obtain 
the  difference  of  latitude  of  the  new  line,  and  thence  calculate 
its  distance  if  desired. 


SEC.  I.] 


LAYING  OUT  LAND. 


277 


§ 


H 


oa 


§ 


28 


3 


oo  oo 

rH    !•* 


278  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

/  Ex.  2.  Required  to  straighten  the  north  boundary  of  the 
tract  the  field-notes  of  which  are  given  Ex.  1,  Art.  389, 
the  new  line  to  run  from  a  point  five  chains  from  the  be- 
ginning of  the  tenth  side.  The  bearing  and  distance  of 
the  new  line,  and  the  position  of  the  point  where  it  strikes 
the  fourth  side,  are  desired. 

Ans.  Division  line,  S.  83°  14'  E.  40.41  chains  to  a  point 
3.51  chains  from  the  beginning  of  the  fourth  side. 

392.  Third  Method. — "When  the  old  lines  do  not  vary 
much  from  the  position  of  the  new,  and  are  crooked,  it  will 
frequently  be  found  most  convenient  to  run  a  "guess-line," 
and  take  offsets  from  this  to  different  points  of  the  bound- 
ary.    Then  calculate  the  contents  of  the  parts  cut  off  on 
each  side  of  this  line.     These,  if  the  assumed  line  were 
correct,  must  be  equal ;   if  they  are  not  so,  divide  the  dif- 
ference of  the  areas  by  half  of  the  length  of  the  "  guess- 
line/'  and  set  the  quotient  off  perpendicular  to  that  line. 
Through  the  extremity  of  that  perpendicular  run  a  parallel 
to  the  "guess-line,"  meeting  the  side  of  the  tract.     The 
division  line  will  run  through  this  point,  very  nearly,  if  the 
"  guess-line"  did  not  differ  much  from  the  true  one.     If 
greater  accuracy  is  required,  the  operation  may  be  repeated, 
using  the  line  determined  by  the  first  approximation  as  the 
basis  of  operations. 

393.  Fourth  Method. — Run  a  random  line  from  the  start- 
ing point  to  the  side  on  which  the  new  line  will  fall,  and 
calculate   the   area   contained  between   this  line  and  the 
original  boundaries.     Then,  by  Art.  378,  run  a  new  line  to 
cut  oft*  the  same  area :  this  will  be  the  line  required. 

Thus,  (Ex.  1,  Art.  390,)  the 
bearing  of  EG  (Fig.  184)  being 
K10}°E:  run  BA  S.  79J° 
E.  45.45  chains,  falling  on  GE 
at  A,  distant  .69  chains  from 
E.  in  GE  produced. 


SEC.  I.] 


LAYING  OUT  LAND. 


279 


a 

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CO 
O 

o 

O  to 
O  b- 

HH   CD 
CD  rH 

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280 


LAYING  OUT  AND  DIVIDING  LAND. 


[CHAP.  VII. 


Problem  12. — To  run  a  new  line  between  two  tracts  of  dif- 
ferent prices,  so  that  the  quantities  cut  off  from  each  may  be  of 
equal  value. 

394.  This  problem  is  in  general  a  very  complicated  one, 
and  can  be  best  solved  by  approximation.     Thus,  run  a 
"guess-line,"  and  calculate  the  area  cut  off  from  each  tract. 
If  these  areas  are  in  the  inverse  ratio  of  the  prices,  the  line 
is  a  correct  one;  if  not,  run  a  new  line  near  this,  and 
repeat  the  calculation :  a  few  judicious  trials  will  locate  the 
line  correctly. 

395.  The  following  cases  admit  of  simple  solutions : — 


CASE  1. 

When  the  old  line  is  straight,  and  the  new  line  is  to  run  a  given 
course. 

The  method  of  solution  will  best  be  shown  by  an  ex- 
ample. 

Let  the  bearings  of  the  Fig.  185. 

lines  be  LA  (Fig.  185)  K 
46°  45'  E.,  AB  S.  71° 
20'  E.,  24.10  chains,  and 
BM  K  10°  35'  E.,  the 
land  to  the  north  of 
AB  being  estimated  at 
$80  per  acre,  and  that  to 
the  south  at  $100  per 
acre.  It  is  required  to 
run  a  new  division  line, 
running  due  east,  so  as 
not  to  alter  the  value  of 
the  two  tracts. 

Through  B  and  A  draw  BD  and  AC  parallel  to  the 
division  line,  and  CF  parallel  to  AB,  meeting  LA  pro- 
duced in  F.  Take  AL  =  \°  AD  =  £  AD,  and  FI  a  mean 
proportional  between  AL  and  AF.  Join  IB,  and  draw  FE 


SEC.  L]  LAYING  OUT  LAND.  281 

parallel  to  it,  meeting  AB  in  E.     Then  the  division  lino 
will  run  through  E. 

DEMONSTRATION.— AL  :  FI  : :  FP :  AF;  .-.  AL  :  AP  : :  FP  :  AF2;  but  AB 
=  |  AL  ;  .-.  AD  :  AF  : :  f  FP  :  AF2  : :  f  BE*  :  AE*  : :  BEa  :  f  AEa. 

But     AD  :  AF  : :  ADB  :  AFB  (1.6)  : :  ADB  :  ABC  :  :  BEa :  f  AEa,       (A) 
and  ABC  :  BEH  ::  ABa :  BEa; 

...  (23.5)  ADB  :  BEH  : :  ABa  :  f  AE« ; 

but  ADB  :  f  AEK :  :  ABa  :  f  AEa,  (Cor.  2,  19.6.) 

.-.BEH  =  f  AEK. 

The  operations  in  the  above  construction  may  readily  be 
done  on  the  ground.  Thus : 

Run  BD,  AC,  and  CF.  Measure  AF  and  AD.  Calcu- 
late </\  AD  .  AF,  which  call  M.  Then  say,  As  AF  +  M 
:  AF  : :  AB  :  AE.  Through  E  run  the  division  line. 

Calculation. 

To  find  AD.  Say,  As  sin.  ADB  (43°  15')  I  sin.  ABD  (18° 
40') : :  AB  (24.10)  :  AD  =  11.26. 

To  find  AF.  Say,  As  sin.  ACB  .  sin.  BAF  :  sin.  BAC  . 
sin.  ABC  : :  AB  :  AF ; 

that  is,  As  sin.  79°  25'  .  sin.  61°  55' :  sin.  18°  40'  .  sin.  81° 
55' : :  24.10  :  AF  =  8.81 ;  FI  =  v'f  AD  .  AF  =  11.13. 

Then,  As  AF  +  FI  (19.94)  :  AF  (8.81)  : :  AB  (24.10) 
:  AE  =  10.64 ; 

Or,  As  AF  +  FI  (19.94)  :  AF  (8.81) : :  AD  (11.26)  :  AK 
=  4.9T. 

CASE  2. 

396.  The  division  line  to  run  through  a  given  point  E  in  AB. 

Let  the  bearings  be  as  in  last  case.  To  run  the  division 
line  through  a  point  E  in  AB  10.64  chains  distant  from  A. 


282 


LAYING  OUT   AND  DIVIDING  LAND. 


[CHAP.  VIL 


Fig.  186. 


Construction. —  Take 
AI  (Fig.  186)  a  third 
proportional  to  BE  and 
AE.  Let  AK  =  f  AI 
and  AL  =  BE.  Draw 
LM  parallel  to  BC, 
cutting  AB  in  1ST ;  and 
KM  parallel  to  AB. 
Make  LO  =  MN. 
Join  AO,  and  draw 
GEH  parallel  to  it. 
Then  the  thing  is  done. 


DEMONSTRATION.  —  Conceive  BC  and  AL  to  meet  in  P.     Then  we  have 

BE  :  EA  :  :  EA  :  AI.     .-.  (Cor.  2,  20.6)  BE  :  AI  :  :  BE*  :  EAa,  and  LA  :  AK 
:  :  BE*  :  £  EAa. 

Again  :  PB  :  PC  :  :  PD  :  PA  :  :  PA  :  PF  :  :  AD  :  AF; 
but          PB  :  PC  :  :  LN  :  LO  :  :  LN  :  NM  :  :  LA  :  AK  :  :  BEa  :  f  EAa; 
whence   AD  :  AF  :  :  BE*  :  f  EA*,  which  agrees  with  (A)  in  the  demonstration 
of  last  case.     Then,  following  the  steps  of  that  demonstration,  we  find  BEH  = 
|  AEG. 

This,  like  the  last  case,  may  readily  be  done  on  the 

AE2 

ground,  thus  ;  Calculate  AI  =  —  —  ,  and  make  AK  =  f  AI. 


Lay  off  on  DA  produced  AL  =  BE  :  run  LETM  and  KM. 
Lay  off  LO  =  NM,  and  run  GEH  parallel  to  AO. 


Calculation. 


AK  = 


5AE2 
4EB 


10.51. 


Then  sin.  M  (81°  55')  :  sin.  AKM  (61°  55')  :  :  AK  (10.51) 
9.37  =  LO  ; 


and,  As  LA  +  LO  (22.83)  :  LA  -  LO  (4.09)  :  :  tan. 


LOA-LAO 


28o  45,. 


LAO  =  71°  55'  -  28°  45'  =  43°  10'. 
But  AP  bears  K  46°  45'  E.  ;  hence  GH  bears  K  89°  55'  E. 


BEG.  L]  LAYING  OUT  LAND.  283 

CASE  3. 

397.  When  the  starting  point  is  in  the  line  AD. 

Given  as  before  to  run  the  line  from  a  point  G  in  AD 
at  4.97  chains  from  A. 

Produce  DA  and  BC  (Fig.  186)  to  meet  in  P.  Calcu- 
late AP  :  let  the  given  ratio  f  be  represented  by  r  :  then, 
As  sin.  P  (36°  10')  :  sin.  ABC  (81°  55')  :  :  AB  (24.10)  :  AP 
=  40.432. 


Put  =  .7636  =  A; 

AJr 

and  M2  =  A  .  PG  =  34.6T. 


Lay  off  GD  =  }  A  ±  V\  A2  +  M2  =  .382  +  5.900  =  6.282, 
(the  lower  sign  being  used  when  G  is  between  A  and  P.) 
Then  GH  parallel  to  DB  will  be  the  division  line. 


DEMONSTRATION.—  Since  GD  =  \  A  -f  </£  Aa  -f  M*, 
we  have  GD  —  \  A  =  ^/\  Am  -f  Ma,  and  GD*  —  A  .  GD  =  M9, 
or  GD  (GD  —  A)  =  A  .  PG  ;  whence  PG  :  DG  :  :  DG  —  A  :  A, 

and  composition,  PD  :  DG  :  :  DG  :  A  (^^T  )  :  :  AP  .  DG  :  r  .  AG»  ; 

whence  r  .  PD  .  AGa  =  AP  .  DGa, 

and      r  .  AGa  :  DG»  :  :  AP  :  PD  :  :  PC  :  PB  :  :  PF  :  PA  :  :  AF  :  AD, 

or,        r  .  AEa  :  EB3  :  :  AF  :  AD.     As  this  agrees  with  (A)  in  the  demonstra- 
tion to  Case  1,  the  truth  of  the  work  is  clear. 

Having  found  AD,  the  bearing  of  DB,  which  is  parallel 
to  GH,  may  be  found  by  calculating  the  angle  ADB  ;  thus  : 
As  (AB  +  AD)  35.352  :  (AB  -  AD)  12.848  :  :  tan. 

ADB-ABD 


Whence  the  angle  ADB  is  43°  157  25",  and  the  bearing  of 
DB  or  GH  is  S.  89°  59'  35/;  E. 

The  whole  of  the  preceding  construction  might  be  made 
geometrically,  but  some  of  the  lines  required  would  be  so 
small  that  no  dependence  could  be  had  on  the  work  ;  the 
method  is  therefore  omitted. 

If  the  given  point  were  not  on  one  of  the  lines,  the  pro- 
blem becomes  very  complicated.  It  may,  however,  be 
solved  by  running  "guess-lines." 


284  LAYING  OUT   AND  DIVIDING  LAND.  [CHAP.  VII. 

SECTION  II. 
DIVISION  OF  LAND. 

Problem  1. — To  divide  a  triangle  into  two  parts  having  a 
given  ratio. 

CASE  1. 

398.  By  a  line  through  one  of  the  corners. 

Divide  the  base  into  two  parts  having  the  same  ratio  as 
the  parts  into  which  the  triangle  is  to  be  divided,  and  draw 
a  line  from  the  point  of  section  to  the  opposite  angle,  (1.6), 

EXAMPLES. 

Ex.  1.  A  triangular  field  ABC  contains  10  acres,  the  base 
AB  being  22.50  chains.  It  is  required  to  cut  off  4J  acres 
towards  the  point  A  by  a  line  CD  from  the  angle  C.  What 
is  the  distance  AD  ? 

Calculation. 
As  10  :  4J  : :  AB  (22.50)  :  AD  =  10.125  chains. 

Ex.  2.  The  area  of  a  triangle  ABC  is  7  acres,  the  side 
AC  being  15  chains.  To  determine  the  distance  AD  to  a 
point  in  AC,  so  that  the  triangle  ABD  may  contain  3  acres. 

Ans.  AD  =  6.43  chains. 

CASE  2. 

399.  JBy  &  line  through  a  given  point  in  one  of  the  sides. 

Say,  As  the  whole  area  is  to  the  area  of  the  part  to  be 
cut  off,  so  is  the  rectangle  of  the  sides  about  the  angle 
towards  which  the  required  part  is  to  lie,  to  a  fourth 
term. 

This  fourth  term  divided  by  the  given  distance  will  give 
the  distance  on  the  other  side. 


SEC.  II.]  DIVISION  OF  LAND.  285 

DEMONSTRATION.— Let  ABC  (Fig.  187)  be  the  given  tri-  Fig.  187. 

angle,  and  ADE  the  part  cut  off.     Then  we  shall  have  C 

(Art.  357)  rad.  :  sin.  A  : :  AB  .  AC  :  2  ABC,  and  rad. 
:  sin.  A  : :  AD  .  AE  :  2  ADE ;  wherefore  2  ABC  :  2  ADE 
: :  AB  .  AC  :  AD  .  AE,  or  ABC  :  ADE  : :  AB  .  AC  :  AD 
.  AE. 


EXAMPLES. 

Ex.  1.  Given  the  side  AB  =  25  chains,  AC  =  20  chains, 
and  the  distance  AD  =  12  chains,  to  find  a  point  E  in 
AB,  such  that  the  triangle  cut  off  by  DE  may  be  to  the 
whole  triangle  as  2  is  to  5. 

Calculation. 
As  5  :  2  : :  AB  .  AC  (500)  :  AD  .  AE  (200) ; 

200 

whence  AE  =  —  =  16.66  chains. 

±Z 

Ex.  2.  Given  AB  =  12.25  chains,  AC  =  10.42  chains,  and 
the  area  of  ABC  =  5  A.  3  E.  8  P.,  to  cut  off  3  acres  to- 
wards the  angle  A  by  a  line  running  through  a  point  E  in 
AB  8.50  chains  from  the  point  A.  Required  the  distance 
on  AC.  Ans.  7.7*7  chains. 

CASE  3. 
400.  By  a  line  parallel  to  one  of  the  sides. 

Since  the  part  cut  off  will  be  similar  to  the  whole,  say, 
As  the  whole  area  is  to  the  area  to  be  cut  off,  so  is  the 
square  of  one  of  the  sides  to  the  square  of  the  correspond- 
ing side  of  the  part. 

The  problem  may  be  constructed  thus : 
Let  ABC  (Fig.  188)  be  the  given  triangle. 
Divide  AB  in  F,  so  that  AF  may  be  to 
FB  in  the  ratio  of  the  parts  into  which 
the  triangle  is  to  be  divided.  Take  AD 
a  mean  proportional  between  AF  and  AB.  Then,  DE 
parallel  to  BC  will  divide  the  triangle  as  required. 

For  AFC  :  FCB  : :  AF  :  FB,  and  (lemma)  ADE  =  AFC ; 
therefore  ADE  :  DECB  : :  AF  :  FB. 


286  LAYING  OUT   AND   DIVIDING  LAND.  [CHAP.  VII. 

EXAMPLES. 

Ex.  1.  The  three  sides  of  a  triangle  are  AB  =  25  chains, 
AC  =  20  chains,  and  BC  =  17  chains,  to  divide  it  into  two 
parts  ADE  and  DECD,  having  the  ratio  of  4  to  3,  by  a  line 
parallel  to  BC. 

Say,  As  7  :  4  : :  AB2  (625)  :  AD2  =  357.1428 ; 
whence  AB  =  18.90  chains. 

Ex.  2.  The  three  sides  of  a  triangle  are  AB  =  25  chains, 
AC  =  20  chains,  and  BC  =  15  chains,  to  divide  it  into  two 
parts  ADE  and  DECB,  which  shall  be  to  each  other  as  2 
to  3,  by  a  line  parallel  to  BC.  "What  is  the  distance  on  AC 
to  the  division  line  ?  Ans.  12.65  chains. 


CASE  4. 
401.  By  a  line  running  a  given  course. 

Construction.— Divide  AB  in  G,  (Fig.  Kg.  m. 

189,)  so  that  AG  may  be  to  GB  in  the 
ratio  of  the  parts  of  the  triangle. 
Run  CF  according  to  the  given  course. 
Take  AD  a  mean  proportional  be- 
tween AF  and  AG.  Then  DE  paral- 
lel to  CF  is  the  division  line. 


w        D          p      x      B 

For  ACG  :  CGB  : :  AG  :  GB,  and,  by  the  lemma,  ADE 

:    ACG. 

ADE  :  DECB  ::  AG:  GB. 

Calculation. 


In  ACF  find  AF.  Then  AD  =  v/AG  .  AF ;  or  say,  As 
the  rectangle  of  the  sines  of  D  and  E  is  to  the  rectangle  of 
the  sines  of  B  and  C,  so  is  the  square  of  BC  to  a  fourth 
term. 

Then,  if  the  ratio  of  the  parts  is  to  be  as  m  to  n,  m  cor- 
responding to  the  triangular  portion,  multiply  this  fourth 
term  by  m,  and  divide  by  m  +  n :  the  quotient  will  be  the 
square  of  DE.  "Whence  AD  is  readily  found. 


SEC.  II.]. 


DIVISION  OF  LAND. 


287 


DEMONSTRATION.  —  Draw  xy  parallel  to  CF,  making  Kxy  =  ABC,  and  draw 
BR  parallel  to  xy.  Then,  as  was  shown  in  Art.  385,  sin.  D  .  sin.  E  :  sin.  B 
.  sin.  C  :  :  BC2  :  zya,  and  (Cor.  2,  20.6)  Axy  :  ADE  or  m  +  n  :  m  :  :  xy*  :  DE" 

EXAMPLES. 

Ex.  1.  The  bearings  and  distances  of  the  sides  of  a  tri- 
angular plat  of  ground  are  AB  N.  71°  E.  17.49  chains,  BC 
S.  15°  W.  12.66  chains,  and  CA  K".  63}°  W.  14.78  chains, 
to  divide  it  into  two  parts  ADE  and  DECB,  in  the  ratio  of 
2  to  3,  by  a  line  running  due  north.  The  distance  AD  is 
required. 

First  Method. 


71° 

63°  45' 
14.78 


As     sin.  F 
:      sin.  ACF 
::    AC 
:      AF 

AG  =  f  AB  =  6.996 


AD  =  9.904  ch. 

Second  Method. 
71° 

63°  45' 
56° 

78°  45' 
12.66 

a 

153.68 

2 


As 


J  sin. 
(sin. 


sin.  D 
E 
sin.  B 
sin.  C 
BC 
BC 
xy* 


As  sin.  A 
:   sin.  E 
::DE 
:  AD 


A.  C.  0.024330 

9.952731 

1.169674 

1.146735 

0.844850 

2)1.991585 

.995792 


A.  C.  0.024330 
0.047269 
9.918574 
9.991574 
1.102434 
1.102434 


2.186615 


DE 


5)307.36 
/  61.472 
45°  15' 
63°  45' 
7.841 
9.902 


7.841. 

A.  C.  0.148628 
9.952731 
0.894371 
0.995730 


288 


LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 


Ex.  2.  Given  AB  K  63°  W.  12.73  ch.,  BC  S.  10°  15'  W. 
8.84  ch.,  and  CA  K  77°  15'  E.  13.24  ch.,  to  determine  the 
distance  AD  on  AB  so  that  DE  perpendicular  to  AB  will 
divide  the  triangle  into  two  equal  parts. 

Ans.  AD  =  8.049  ch. 


Fig.  190. 


CASE  6. 

402.  By  a  line  through  a  given  point. 

Let  ABC  (Fig.  190)  be  the  tri- 
angle to  be  divided  into  two  parts 
CLK  and  ABKL,  which  shall  be 
to  each  other  as  the  numbers  m 
and  n  :  the  division  line  to  run 
through  a  given  point  P. 

Construction. 


Bisect  BC  in  D;  divide  CA  in  F,  so  that  CF  :  FA  : :  m  : 
n.  Through  P  draw  HPE  parallel  to  BC.  Join  ED ;  draw 
FG-  parallel  to  it,  and  complete  the  parallelogram  CH. 
Make  GI  perpendicular  to  BC  and  equal  to  EP.  "With  the 
centre  I  and  the  radius  PH,  describe  an  arc  cutting  BC  in 
K ;  then  KPL  will  be  the  division  line. 

If  IG  is  greater  than  IK,  the  question  is  impossible  in  the 
terms  proposed.  The  triangular  part  will  then  be  adjacent 
to  one  of  the  other  angular  points,  and  a  construction  alto- 
gether analogous  to  the  above  will  fix  the  position  of  the 
division  line. 

DEMONSTRATION. — Conceive  DA,  DF,  and  EG  to  be  joined.  Then,  since  CD  = 
$  BC,  ADC  =  £  ABC,  and,  because  CF  :  FA : :  m  :  n,  we  have  by  composition 


CA  :  CF  : :  m  -f-  n  :  m;  whence  CFD  = 


r»-J-n 


CAD.    But  CDF  =  CEG,  and  CH 


2  CEG  .-.  CH  = CAB,  and  by  demonstration  (Art.  381)  CKL  =  CH  ; 

m  -}-  n 


therefore  CKL 


CAB. 


Sic.  II.  J  DIVISION  OF  LAND.  289 

Calculation. 

Find  PE,  EC,  and  FC  =  — ^—  AC ;  then  CE  :  CF  : :  CD 

m  +  n 


(|  EC)  :  CG,  and  KG  =  v/  KP  -  IG2  =  V  PH2  —  PE2. 
Finally,  CK  =  CG  ±  GK 

EXAMPLES. 

Ex.  1.  Given  the  bearings  and  distances  of  the  adjacent 
sides  of  a  triangular  tract,— viz. :  CA  K  10°  17'  W.  13.25 
ch.,  CB  N".  82°  5'  W.  13.75  ch.,— to  divide  it  into  two  por- 
tions ABKL  and  KLC  in  the  ratio  of  4  to  5,  by  a  line  through 
a  point  P  N.  28  W.  7.85  chains  from  the  corner  C.  The 
distance  CK  is  required. 

Calculation. 
To  find  PE  and  EC. 

As  sin.  PEC                 108°  12'  A.  C.  0.022289 

:    sin.  PCE                 17°  43'  9.483316 

::  PC                           7.85  0.894870 

:    PE                           2.515  0.400475 

As  sin.  PEC                 108°  12'  A.  C.  0.022289 

:    sin.  CPE  ..,,,    54°  5'  9.908416 

::  PC  0.894870 

:    CE                           6.692  0.825575 

To  find  CG. 

As  CE  6.692                    A.  C.  9.174425 

:    CF  =  f  CA  7.361                               0.866937 

: :  CD  =  |  CB  6.875                               0.837273 

:   CG  =  EH  7.562                              0.878635 

EP  2.515 

PH  =  IK  =  5.047 

19 


290  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

To  find  KG  and  CK. 

KI  +  IG  7.562  0.878635 

KI  -  IG  2.532  0.403464 

2)1.282099 

KG  =  4.376  .641049 

CG  = 
CK  = 

Ex.  2.  Given  AB  K.  46°  15'  E.  8.80  ch.,  AC  S.  65°  15'  E. 
11.87  ch.,  to  determine  the  distance  AK  to  a  point  K  in  AB 
so  that  a  line  from  K  through  a  spring  P  K  80°  E.  5.90  ch. 
from  A  may  divide  the  triangle  into  two  equal  parts. 

Ans.  AK  =  8.58  ch.,  or  6.244  ch. 

Problem  2.  To  divide  a  trapezoid  into  two  parts  having  a 
given  ratio. 

CASE  1. 

403.  By  a  line  cutting  the  parallel  sides. 

Fig.  191. 

a.  Divide  DC  and  AB  (Fig.  191)  »J* * 

in  F  and  E  so  that  the  parts  may  have 

the  same  ratio  as  the  parts  into  which 
the  trapezoid  is  to  be  divided:  join 
EF  and  the  thing  is  done. 

b.  If  the  division  line  is  to  pass  through  a  given  point  G 
in  one  of  the  parallel  sides.     Determine  F  and  E  as  before ; 
then  lay  off  EH  =  FG,  and  GH  will  be  the  division  line. 

c.  If  the  division  line  is  to  pass       D  G     Fjp  192-        c 
through  a  point  P  (Fig.  192)  not  in 

AB  or  CD.  Determine  EF  as 
before.  Bisect  it  in  I.  Through  P 
and  I  draw  the  division  line  GH. 

Should  GH  cut  either  of  the  non-  A 
parallel  sides  before  it  does  both  of  these,  one  of  the  por- 
tions will  be  a  triangle.  It  will  then  be  necessary  to  calcu- 
late the  area  of  the  whole  tract,  whence  that  of  each  por- 
tion is  found.  Then,  by  Art.  381,  lay  off  a  triangle  by  a  line 
through  P  so  as  to  contain  the  required  area. 


SEC.  II.]'  DIVISION  OF  LAND.  291 

Calculation. 

Through  P  draw  MPL  parallel  to  AB,  and  from  the  data 
given  find  AM  and  MP. 

Then  DA  :  AM  : :  AE  —  DF  :  AE  -  LM;  whence  LM 
and  PL  are  known. 

But  AM  —  J  AD  :  J  AD  : :  PL  :  GF  =  EH;  and  DG  = 
DF  -  FG. 

EXAMPLES. 

Ex.  1.  Given  AB  E.  9.10  ch.,  BC  K  14°  20'  W.  4.40  ch., 
CD  W.  6.95  ch.,  and  DA  S.  14°  W.  4.39  ch.,  to  divide  the 
tract  into  two  parts  having  a  ratio  of  3  to  4  by  a  line  HG 
through  a  spring  N.  47°  E.  4.40  ch.  from  the  corner  A;  the 
smaller  division  to  be  next  to  AD.  Eequired  the  distances 
of  the  division  line  from  A  and  D. 

Calculation. 

To  find  AM  and  MP. 

As  sin.  M                       76°  A.  C.  0.013096 

:    sin.  APM                 43°  9.833783 

::  AP                            4.40  0.643453 

:    AM                           3.093  0.490332 

And  As  sin.  M  A.  C.  0.013096 

:   sin.  PAM  33°                                  9.736109 

::  AP  0.643453 

:  PM  2.470                              0.392658 

To  find  EH,  AH,  and  DG. 

DF  =  f  DC  =  2.979,  and  AE  =  f  AB  =  3.90. 
Then,  As  AD  (4.39)  :  AM  (3.093)  : :  AE  —  DF  (.921)  :  Ah 
—  ML  =  .649; 

whence  ML  =  3.251,  and  PL  =  3.251  —  2.470  =  .781. 
As  AM  —  I  AD  (.898)  :  J  AD  (2.195)  : :  PL  (.781) :  FG  = 
EH  =  1.909.  Finally,  AH  =  AE  +  EH  =  5.81,  and  DG 
=  DF  —  FG  =  1.07. 

Ex.  2.  Given  AB  S.  62°  50'  E.  14.93  ch.,  BC  K  7°  30'  W. 
6.29  ch.,  CD  K  62°  50'  W.  11.88  ch.,  DA  S.  21  W.  5.18  ch., 


292  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

to  determine  DG  and  AH  so  that  a  line  joining  G  and  H 
will  pass  through  P  K  75°  50'  E.  6.20  ch.  from  A,  and  cut 
off  one-third  of  the  area  of  the  tract  towards  AD. 

Ans.  AH  =  3.40  ch. ;  DG  =  5.53  ch. 

CASE  2. 

404.  The  division  line  to  be  parallel  to  the  parallel  sides. 

Fig.  193. 


Let  ABOD  (Fig.  193)  be  the  trape- 
zoid  to  be  divided  into  two  parts  AEFD 
and  FEBC  having  the  ratio  of  two 
numbers  m  and  n  by  a  line  EF  parallel 
to  AD  or  EC. 


Construction. 

Join  CA,  and  draw  DH  parallel  to  it.  Join  CH.  Divide 
HB  in  I  so  that  HI :  IB  : :  m  :  n.  Produce  CD  and  BA  to 
meet  in  G,  and  take  GE  a  mean  proportional  between  GI 
and  GB.  Join  CI,  and  draw  EF  parallel  to  AD :  then  will 
EF  be  the  division  line  required. 

DEMONSTRATION.— Because  DH  is  parallel  to  CA,  AHC  =  ADC  (37.1);  .-. 
ABCD  =  BCH,  and,  since  HB  is  divided  in  I  so  that  HI  :  IB  : :  m  :  n,  we  have 
CHI  :  GIB  : :  m  :  n  (1.6.)  These  triangles  are  therefore  equal  to  the  parts  into 
which  the  trapezoid  is  to  be  divided.  But  (lemma)  GEF  =  GIC :  therefore 
EBCF  =  ICB,  and  £F  is  the  division  line. 

Calculation. 

EF  may  be  found  by  the  formula  EF2  =  ^_BC2  +  n  AI)2 

m  +  n 
then   BC  <AS>  AD  :  EF  sss  AD  : :  AB  :  AE. 

DEMONSTRATION.— GBC  :  GAD  ::  BCa  :  AD»;  .-.  (17.5)  ABCD  :  GAD  :: 
BCa  —  AD'  :  AD*. 

Similarly,  GEF  :  GAD ::  EFa:  AD9  .-.  (17.5)  AEFD:  GAD  : :  FEa  —  AD*:  AD*; 
whence  ABCD  :  AEFD  : :  BC3  —  ADa  •  FEa  —  ADa; 

or,  m  +  n  :  m  : :  BCa  —  ADa  :  FEa  —  ADa : 

consequently  (m  -f-  n)  FEa  —  m  ADa  —  n  ADa  =  m  BCa  —  m  ADa; 

m  BCa  +  n  ADft 
or,  (m  +  n)  FEa  =  m  BCa  +  n  AD2,  and  FEa  = — . 

in  +  n 

Again :  Draw  AKL  parallel  to  DC.     Then  BL  :  EK  : :  AB  :  AE ;     . 
or,  BC  —  AD  :  FE  —  AD  :  :  AB  :  AE. 


SEC.  II.]  DIVISION  OF  LAND.  293 

Second  Method. 
The  distance  AE  may  be  calculated  thus  :— 

Find  GA  and  GD;  thence  GO  and  GB  are  known: 

then  GC  :  GD  : :  GA  :  GH;  whence  HB  and  HI  are  known, 

and  therefore  GE  =  */  GI.GB  is  known. 

EXAMPLES. 

Ex.  1.  Given  AB  S.  14°  W.  4.39  ch.,  BC  E.  9.10  ch.,  CD 
K  14°  20'  W.  4.40  chains,  and  DA  W.  6.95  chains,  to  divide 
the  trapezoid  into  two  parts  AEFD  and  BEFC  in  the  ratio 
of  2  to  3,  by  a  line  EF  parallel  to  the  sides  BC  and  DA. 
Kequired  the  distance  AE  on  the  first  side. 

m .  BC2  +  n .  AD2      165.62  +  144.9075 

• = 

m  +  n  5 


whence  EF  =  v'  62.1055  =  7.88. 

And  BC  -  AD  (2.15)  :  EF  -  AD  (.93)  : :  AB  (4.39)  :  AE 
=  1.90. 

Ex.  2.  Given  AB  S.  87°  15'  E.  6.47  chains,  BC  N.  23° 
30'  E.  10.32  chains,  CD  S.  64°  45'  W.  9.30  chains,  and  DA 
S.  23°  30'  W.  5.55  chains,  to  determine  the  distance  AE  of 
a  point  E,  situated  in  AB,  such  that  EF  parallel  to  AD 
may  divide  the  trapezoid  into  two  parts  AEFD  and 
EBCF  having  the  ratio  of  4  to  5. 

Ans.   AE  =  3.36  chains. 


294  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.VIL 

Problem  3. — To  divide  a  trapezium  into  two  parts  having  a 
given  ratio. 

CASE  1. 

405.  The  division  line  to  run  through  a  given  point  in  one  of 
the  sides. 

Let  ABCD  (Fig.  194)  represent 
the  trapezium  and  P  the  given 
point ;  and  let  m  :  n  represent  the 
given  ratio. 

CONSTRUCTION. — Determine  I,  as 
in  Art.  404.     Join  PI,  and  draw   G 
CF  parallel  to  it :  then  will  PF  be  the  division  line. 

For  if  CH  and  CI  be  joined,  CHD  =  ABCD ;  and,  since 
HCI :  ICD  : :  m  :  n,  HCI  and  ICD  will  be  equal  to  the  two 
parts  into  which  the  quadrilateral  is  to  be  divided.  But, 
since  PI  is  parallel  to  CF,  we  have 

GC :  GP : :  GF :  GI;  .-.  (15.6)  GPF  =  GCI,  andPFDC  =  CID. 

Calculation. 

In  GAB  find  GA  and  GB. 
Then  GC  :  GB  : :  G  A  :  GH; 

whence  HD  and  HI  become  known ; 

and  GP  :  GC  : :  GI  :  GF. 

Finally,  AF  =  GF  -  GA. 

EXAMPLES. 

Ex.  1.  Given  AB  K  25|°  E.  4.65  chains,  BC  K  77°  E. 
6.30  chains,  CD  South  7.30  chains,  and  DA  K  78J°  W. 
8.35  chains,  to  divide  the  trapezium  into  two  equal  parts  by 
a  line  EF  running  through  a  point  P  in  BC  distant  2.50 
chains  from  B.  AF  is  required. 


SBO.  II.]  DIVISION  OF  LAND.  295 

Calculation. 

To  find  GA  and  GB. 

As  sin.  G  24°  45'  A.  C.  0.378139 

:     sin.  GBA  51°  15'  9.892030 

:  :  AB  4.65  0.667453 

:    AG  8.662  0.937622 

AD  8.35 

GD  17.012 

As  sin.  a  24°  45'  A.  C.  0.378139 

:  sin.  GAB  104°  9.986904 

::  AB  0.667453 

:  BG  10.777  1.032496 

BC 

GO 

To  find  GH. 

As  GO  17.077  A.  C.  8.767588 

:    GB  10.777  1.032496 

:  :  GA  8.662  0.937622 

:     GH  5.466  0.737706 

HI  =  J  (GD  -  GH)  =  5.773  and  GI  =  GH  +  HI  =  11.239, 

To  find  GF  and  AF. 
AsGP 
:    GC 
::   GI 
:    GF 
AG 
AF  5.794. 

Ex.  2.  Given  AB  K  27|°  W.  19.55  chains,  BC  East 
18.92  chains,  CD  S.  1J°  E.  10.49  chains,  and  DA  S.  56°  W. 
12.25  chains,  to  find  BF,  so  that  a  line  run  from  a  point 


13.277 

A.  C.  8.876900 

17.077 

1.232412 

11.239 

1.050727 

14.456 

1.160039 

8.662 

296  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII 

P  in  AD  6  chains  from  A  may  divide  the  trapezium  into 
two  parts  ABFP  and  PFCD  having  the  ratio  of  5  to  4. 

Ans.   BF  =  9.00  ch. 

CASE  2. 
406.   The  division  line  to  run  through  any  point. 

Let  ABCD  (Fig.  195)  Fig.  195. 

"be  the  given  trapezium 
and  P  the  given  point. 

Determine  I,  as  in  the  G^---~        ,  ^  y       -«         — __  M 
last  two  articles,  and  bi- 
sect GI  in  K    Through 
P  draw  0PM  parallel  to 
GD,  meeting  GB  in  0.  }/ 

Join  EX),  and  draw  CL  N 

parallel  to  it.    Through 

L  draw  LM  parallel  to  GB.  Make  ~LN  perpendicular  to 
AD  and  equal  to  OP.  With  the  centre  N"  and  radius 
equal  to  PM,  describe  an  arc  cutting  AD  in  F.  Then  FPE 
will  be  the  division  line. 

DEMONSTRATION. — As  was  proven,  Art.  381,  GFE  =  GOML  =  2  GOL  = 
2  GCK  =  GCI :  whence  ABEF  =  ABCI.  But  CI  divides  the  trapezium  into 
two  parts  having  the  given  ratio ;  therefore,  EF  does  so  likewise. 

Calculation. 

Find  GB,  GA,  GH,  and  GI.  Then  in  OBP  find  OB  and 
OP:  thus  GO  is  known.  And  because  GO  :  GO  : :  GK  : 
GL,  GL  is  known ;  but  PM  =  GL  —  OP.  Hence,  in  LNF 
we  have  LN  and  NF  to  find  LF, 

EXAMPLES. 

Ex.  1.  Given  AB  K  25| °  E.  4.65  chains,  BC  N".  77°  E. 
6.30  chains,  GD  South  7.30  chains,  and  DA  N.  78J°  W.  8.35 
chains,  to  part  off  two-fifths  of  the  tract  next  to  AB  by  a 
line  through  a  spring  S.  54|°  E.  2.95  chains  from  the  second 
corner.  The  distance  AF  is  required. 


SEC.  II.] 


DIVISION  OF  LAND. 

Calculation. 


297 


As  in  Ex.  1,  last  case :  GB  =  10.77T,  GA  =  8.662,  GO 
=  17.077,  GD  =  17.012,  GH  =  5.466,  GI  =  (GH  +  f  HD) 
=  10.084,  and  GK  =  5.042. 


As  sin.  BOP 
:     sin.  BPO 
::  BP 
:     OB 
GB 
GO 

As  sin.  BOP 
:     sin.  OBP 
::  BP 
:     OP 


As  GO 
:  GO 
::  GK 
:  GL 


To  find  OB  and  OP. 

24°  45'         A.  C.  0.378139 
9.600700 

0.469822 


24°  45' 
131°  45' 

5.257 

To  find  GL. 

7.967 
17.077 

5.042 
10.807 


0.448661 


A.  C.  0.378139 

9.872772 
0.469822 
0.720733 


9.098705 
1.232412 
0.702603 
1.033720 


KF  =  GL  -  OP 

Whence          LF  =  ^ETF2-LK3 
whence      AF  =  GL  +  LF  —  GA 


5.55. 

1.779; 

3.924. 


Ex.  2.  Given  AB  K  27J°  W.  19.55  chains,  BC  East 
18.92  chains,  CD  S.  1}°  E.  10.49  chains,  and  DA  S.  56° 
W.  12.25  chains,  to  divide  the  quadrilateral  into  two  parts 
ABEF  and  FECD  in  the  ratio  of  5  to  4,  by  a  line  EF 
through  a  spring  P,  which  bears  from  B  S.  70|°  E.  11.52 
chains.  The  distance  AF  is  required. 

Ans.   AF  =  5.01  ch. 


298 


LAYING  OUT  AND  DIVIDING  LAND. 


[CHAP.  VII. 


CASE  3. 

407.   The  division  line  to  be  parallel  to  one  side. 


Fig.  196. 


Let  ABCD  (Fig.  196)  re- 
present the  trapezium  which 
is  to  be  divided  into  two 
parts  having  the  ratio  of  m 
to  n  by  a  line  parallel  to  CD. 


CONSTRUCTION.  —  Deter- 
mine H  and  I,  as  in  the  pre- 
ceding articles.     Take  GF  a  mean  proportional  between 
GI  and  GD :  then  EF,  parallel  to  CD,  will  be  the  division 
line. 
For,  as  was  demonstrated,  (Art.  404,) 

ABCD  =  HCD, 

and  CHI  :  CID  ::  m  :  n. 

But  (lemma)  GCI  =  GEF ; 

ICD  =  EFDC, 

and  HCI  =  ABEF: 

whence  ABEF  :  FECD  : :  m  :  n. 

If  the   division  line  Mg.W. 

is  to  be  parallel  to  the 
shorter  side  AB  (Fig. 
197.)  Draw  CK  paral- 
lel to  AB,  and  take  GF 
a  mean  proportional 
between  GI  and  GK; 
or,  join  BD,  and  draw 
CH'  parallel  to  it. 

AI'  :  I'H'  : :  m  :  n, 

and  take  GF  a  mean  proportional  between  GA  and  GF. 
Then  will  EF,  parallel  to  AB,  be  the  division  line. 


H  A     I      F      K 

Divide  AH'  in  I',  so  that 


SEC.  II.].  DIVISION  OF  LAND.  299 

Calculation. 

First  Method.—  Find,  as  in  the  preceding  articles,  GH 
and  GI.  Then  GF  =  </GI.GD,  or=  -v/GLGK. 

Second  Method.—  Draw  xy  (Fig.  196)  parallel  to  EF,  so  as 
to  make  Gxy  =  GAB,  or  Gxy  =  GOD,  (Fig.  197.)  Then  we 
shall  have 

sin.  E  .  sin.  F  :  sin.  A  .  sin.  B  :  :  AB2  :  xy2,  (Fig.  196,) 
or   sin.  E  .  sin.  F  :  sin.  C  .  sin.  D  :  :  CD2  :  xf\  (Fig.  197  ;) 


and  (Art.  404)  EF2  =      '  '        (Fig.  196  ;) 

m  +  n 


EP=. 

m  -f  n 

DEMONSTRATION.—  Draw  AM   and  BN  (Fig.  196)  parallel  to  EF. 
Then  sin.  M  .  (sin.  E)  :  sin.  B  :  :  AB  :  AM, 

and  sin.  N  .  (sin.  F)  :  sin.  A  :  :  AB  :  BN  ; 

(23.6)  sin.  E  .sin.  F  :  sin.  A  .  sin.  B  :  :  ABa  :  AM  .  BN. 

Now,  since  Gxy  =  GAB,  Gz  is  a  mean  proportional  between  GA  and  GN. 
Wherefore  xy  is  a  mean  proportional  between  AM  and  BN.  Hence,  AM  .  BN 
=  ^; 

consequently,  sin.  E  .  sin.  F  :  sin.  A  .  sin.  B  :  :  ABa  :  xy9. 

If  EF  is  parallel  to  AB,  (Fig.  197,)  the  demonstration  will  be  precisely  similar 
to  the  above. 

EXAMPLES. 

Ex.  1.  Given  the  bearings  and  distances  as  follow,  —  viz.  : 
AB  K  25}°  E.  4.65  chains,  BC  K  77°  E.  6.30  chains,  CD 
South  7.30  chains,  and  DA  K  78J°  W.  8.35  chains,—  to 
divide  the  trapezium  into  two  parts  ABEF  and  FECD, 
having  the  ratio  of  2  to  3,  by  a  line  EF  parallel  to  AB. 
AF  and  EF  are  required. 

Calculation. 

First  Method.—  As  in  Ex.  1  of  Art.  405,  we  find  GA  = 
8,662,  GB  =  10.777,  GO  =  17.077,  GD  =  17.012,  GH  = 
5.466,  and  GI  =  GH  +  f  HD  =  10.084. 


300  LAYING  OUT  AND  DIVIDING  LAND.  [CHAP.  VII. 

To  find  GK  and  GF. 

AsGB-  10.777  A.  C.  8.967504 

:     GA  8.662  0.937622 

:  :  GO  17.077  1.232412 

:     GK  1.137538 

GI  10.084  1.003633 

2)2.141171 


GF  =  >/  GI .  GK  =  11.765  1.070585 

8.662 


AF  =  3.103 

To  find  EF. 

As  GA                          8.662  A.  C.  9.062378 

:    AB                            4.65  0.667453 

:  :  GF                          11.765  1.070585 

:    EF                            6.316  1.800416 

Second  Method. 

C  sin.  E 

As 


sin.  E 

128°  45' 

A.  C.  0.107970 

sin.  F 

76° 

«  «  0.013096 

sin.  C 

77° 

9.988724 

sin.  D 

78°  15' 

9.990803 

CD 

7.30 

0.863323 

CD 

• 

0.863323 

f 

67.18 

1.827239 

2 

134.36 

AB2 

64.8675 

5)199.2275 
EF  =  V  39.8455  =  6.312. 


To  find  AF. 

As  sin.  G 

24°  45' 

A.  C.  0.378139 

:    sin.  E 

128°  45' 

9.892030 

:  :  FE  -  AB 

1.662 

0.220631 

:    AF 

3.096 

0.490800 

SEC.  II.]- 


DIVISION  OP  LAND. 


301 


Ex.  2.  Given  the  bearings  and  distances  as  in  Ex.  1, 
to  divide  the  trapezium  into  two  parts  AFED  and  FECB, 
having  the  ratio  of  3  to  2,  by  a  line  EF  parallel  to  BC.  AF 
and  EF  are  required. 

Ans.  AF  =  1.60  chains ;  EF  =  7.66  chains. 

Ex.  3.  Given  as  in  Ex.  1,  to  divide  the  trapezium  into 
two  parts  ABEF  and  FECD,  in  the  ratio  of  2  to  3,  by  a 
line  EF  parallel  to  CD.  AF  and  EF  are  required. 

Ans.  AF  =  4.44  chains ;  EF  =  5.62  chains. 

CASE  4. 
408.  The  division  line  to  run  any  direction. 


Fig.  198. 


Kv    D 


Let  ABCD  (Fig.  198)  be 
the  trapezium  to  be  divided 
into  two  parts  ABEF  and 
FECD,  in  the  ratio  of  m  to 
ft,  by  a  line  EF  running  any 
course. 

The  construction  of  this 
case  is  the  same  as  that  of 
the  last, — CK  being  drawn  so  as  to  be  of  the  same  course  as 
EF. 

Calculation. 

Conceive  xy  and  vw  to  be  drawn  so  as  to  make  Gxy  = 
GAB,  and  Gvw  =  GCD  :  then  will  vwyx  be  equal  to  ABCD. 
It  will  also  be  divided  by  EF  into  two  parts  having  the 
ratio  of  m  to  n. 

Find  xy2  and  viv3  by  the  proportions 

sin.  E  .  sin.  F  :  sin.  A  .  sin.  B  : :  AB2 :  xy\ 
and        sin.  E  .  sin.  F  :  sin.  C  .  sin.  D  : :  CD3 :  viv2, 

the  truth  of  which  has  been  proven  in  the  demonstration  to 
rule  for  Art.  407. 

Then  (Art.  404)  EF*  =  >»•»*  +  *•&. 

m  +  n 

Draw  AOP  parallel  to  BC,  meeting  BE"  and  EF  in  O 
and  P. 


302  LAYING   OUT   AND    DIVIDING   LAND.          [CHAP.  VII. 

Then  sin.  BOA  (sin.  E) :  sin.  BAG  (sin.  B)  : :  AB  :  BO, 
and  sin.  PAF  (sin.  G)  :  sin.  P  (sin.  E)  : :  PF  (EF  —  BO) 
:  AF. 

The  calculation  may  otherwise  be  made  by  finding  GH 
and  GI,  as  in  Arts.  406,  407,  and  also  GK.  Then  GF  = 

</GI .  GK. 

EXAMPLE. 

Ex.  1.  The  bearings  and  distances  being  as  in  the  ex- 
amples in  last  case,  it  is  required  to  divide  the  trapezium 
into  two  parts  ABEF  and  FECD,  having  the  ratio  of  2  to 
3,  by  a  line  perpendicular  to  AD.  To  find  AF  and  EF. 

Ans.  AF  =  3.84;  EF  =  5.76. 


CHAPTER  VIII. 

MISCELLANEOUS    EXAMPLES. 

Ex.  1.   Two  sides  of  a  triangle  are  32  and  50 
respectively.     Eequired  the  third  side,  so  that  the  area 
may  be  3  acres.  Ans.  31.05  P.  or  78  P. 

Ex.  2.  A  gentleman  has  a  garden  in  the  form  of  a  rect- 
angle, the  adjacent  sides  being  120  and  100  yards  respec- 
tively. There  is  a  walk  half  round  the  garden,  which 
takes  up  one-eighth  of  the  ground.  What  is  its  width  ? 

Ans.  7.05  yards. 

Ex.  3.  The  three  sides  of  a  triangle  are  in  the  ratio  of 
the  numbers  3,  4,  and  5.  What  are  their  lengths,  the  area 
being  2  A.,  IK,  24  P.? 

Ans.  6  chains,  8  chains,  and  10  chains. 

Ex.  4.  The  diameter  of  a  circular  grass-plat  is  150  feet, 
and  the  area  of  the  walk  that  surrounds  it  is  one-fourth  of 
that  of  the  plat.  Required  the  width. 

Ans.  8.85  feet. 

Ex.  5.  To  determine  the  height  of  a  liberty-pole  which 
had  been  inclined  by  a  blast  of  wind,  I  measured  75  feet 
from  its  base,  the  ground  being  level,  and  took  the  angle 
of  elevation  of  its  top  67°  43'  30",  the  angle  of  position 
of  the  base  and  top  being  5°  37'.  Then,  measuring  100 
feet  farther,  I  found  the  angle  of  position  of  the  bottom 
and  top  to  be  2°  29'.  Eequired  the  length  of  the  pole. 

Ans.  194  feet. 

Ex.  6.  The  distances  from  the  three  corners  of  a  field  in 
the  form  of  an  equilateral  triangle  to  a  well  situated  within 
it  are  5.62  chains,  6.23  chains,  and  4.95  chains  respectively. 
What  is  the  area  ?  Ans.  4  A.,  0  R.,  6  P. 

303 


304  MISCELLANEOUS  EXAMPLES.  [CHAP.  VIII. 

Ex.  7.  At  a  station  on  the  side  of  a  pond,  elevated  30 
feet  above  the  water,  the  elevation  of  the  summit  of  a  cliff 
on  the  opposite  shore  was  found  to  be  37°  43'  and  the  de- 
pression of  the  image  45°  26'.  Eequired  the  elevation  of 
the  cliff.  Ans.  221.8ft. 

Ex.  8.  To  find  the  altitude  of  a  tower  on  the  brow  of  a 
hill,  I  measured,  on  slightly-inclined  ground,  a  base-line 
AB  157  yards,  A  being  on  a  level  with  the  base  of  the 
hill.  At  A  the  angle  of  position  of  B  and  C  was  87° 
45';  elevation  of  B,  2°  17';  of  base  of  tower,  39°  43',  and 
of  top,  52°  13'.  At  B  the  depression  of  A  was  2°  17';  the 
angle  of  position  of  A  and  C,  54°  23' ;  elevation  of  base 
of  tower,  33°  4',  and  of  top,  45°  42'.  Kequired  the  height 
of  the  hill  and  also  of  the  tower. 

Ans.  Height  of  hill,  172.5  ft.;  of  tower,  95.5  ft. 

Ex.  9.  To  determine  the  height  of  a  tree  C  standing  on 
the  opposite  shore  of  a  river,  I  measured  a  base-line  AB  of 
100  feet.  At  A  the  angle  BAG  was  90°,  and  the  angle 
of  depression  of  the  image  of  the  top  of  the  tree  was  39° 
48'.  At  B  the  angle  of  depression  was  32°.  Eequired  the 
height,  the  instrument  having  been  10  feet  above  the  water 
at  each  station.  Ans.  84.47  feet. 

Ex.  10.  ~Not  being  able  to  measure  directly  the  three  sides 
of  a  triangle,  the  corners  of  which  were  visible  from  each 
other,  I  took  the  angles  as  follow, — viz. :  A  =  57°  29', 
B  =  72°  41',  and  C  =  49°  50'.  I  also  measured  the  dis- 
tances from  the  corners  to  a  point  within  the  triangle,  and 
found  them  to  be  AD  =  7.56  chains,  BD  =  9.43  chains, 
and  CD  =  8.42  chains.  Eequired  the  lengths  of  the  sides. 
Ans.  AB  =  12.63  chains,  AC  =  15.78  chains,  and  BC 
=*  13.94  chains. 

Ex.  11.  The  base  of  a  triangle  being  50  perches,  and  the 
area  5  acres,  what  are  the  other  sides,  their  sum  being  85 
perches  ?  Ans.  33.3785  P.  and  51.6215  P. 

Ex.  12.  It  is  required  to  lay  out  7  acres  in  a  triangular 
form,  one  side  being  20  chains,  and  the  others  in  the  ratio 
of  2  to  3. 


MISCELLANEOUS  EXAMPLES.  305 

Ans.  The  other  sides  are  9.86   and   14.79   chains,  or 
39.58  and  59.3T  chains. 

Ex.  13.  The  bearings  of  the  dividing  lines  of  two  farms 
being  as  follow,— viz. :  1.  K  83J°  E.  2.3T  chains;  2.  S.  47° 
E.  6.25  chains ;  3.  K  62}°  E.  5.17  chains ;  4.  S.  56J°  E. 
3.92  chains,  and  5.  N.  14J°  E., — it  is  required  to  straighten 
the  boundary,  the  new  line  to  start  from  the  beginning  of 
the  first  side  and  fall  on  the  last.  The  bearing  of  the  new 
line  is  required,,  and  also  the  distance  on  the  last  side. 

Ans.  Bearing,  S.  74°  40'  E.  to  a  point  .25  chains  back 
from  the  commencement  of  the  last  side. 

Ex.  14.  One  side  of  a  tract  running  through  a  thick 
copse,  I  took  a  station  S.  26J°  E.  1.53  chains  from  the 
corner,  and  ran  a  "guess-line"  bearing  N.  60 J°  E.  19.37 
chains,  when  the  other  end  bore  N.  28 J°  W.  3.27  chains. 
What  is  the  course  and  distance  of  the  line,  and  what  must 
be  the  course  and  distance  of  an  offset  from  a  point  8.53 
chains  on  the  random  line,  that  it  may  strike  a  stone  in  the 
side  8.53  chains  from  the  point  of  beginning? 

Ans.  Side,  K  55°  22'  E.  19.42  chains ; 
Offset,  K  28°  8'  "W.  2.29  chains. 

Ex.  15.  Three  observers,  A,  B,  and  C,  whose  distances 
asunder  are  AB  =  1000  yards,  BC  =  1180  yards,  and  AC 
=  1690  yards,  take  the  altitude  of  a  balloon  at  the  same 
instant,  and  find  it  to  be  as  follow, — viz. :  At  A,  53°  43', 
at  B,  46°  40',  and  at  C,  52°  46'.  Eequired  the  height  of 
the  balloon.  Ans.  1461.4  yards  or  2411  yards. 

Ex.  16.  The  bearings  and  distances  of  the  sides  of  a  tract 
of  land  are,— 1.  K  61°  20'  W.  22.55  chains;  2.  K  10°  W. 
16.05  chains ;  3.  K  60°  45'  E.  14.30  chains ;  4.  S.  66°  40'  E. 
17.03  chains ;  5.  S.  86°  E.  22.40  chains ;  6.  S.  31°  40'  E. 
19.10  chains,  and  7.  S.  76°  35'  W.  39  chains,— to  divide  it 
into  two  equal  parts  by  a  line  running  due  north.  The 
position  of  the  division  line  is  desired. 

Ans.  The  division  line  runs  from  a  point  on  the  7th 
side  3.77  chains  from  the  end  thereof. 

20 


306  MISCELLANEOUS   EXAMPLES.  [CHAP.  VIII. 

Ex.  17.  Not  being  able  to  run  a  line  directly,  on  account 
of  a  projecting  cliff,  I  took  the  angles  of  deflection  and  the 
distances  as  follow, — viz. :  1.  to  the  right,  67°  35'  10  chains ; 
2.  to  the  left,  48°  43'  7.25  chains;  3.  to  the  left,  11°  45' 
5.43  chains,  and  4.  to  the  left,  65°  17'.  How  far  on  the  last 
course  must  I  run  before  coming  in  line  again?  at  what 
angle  must  I  deflect  to  continue  the  former  direction  ?  and 
what  is  the  distance  on  the  first  line  ? 

Ans.  Distance  on  the  last  course,  14.42  chains;  on 
the  first,  23.67  chains ;  deflection,  58°  10'  to  the  right. 

Ex.  18.  To  find  the  length  of  a  tree  leaning  to  the  south, 
I  measured  due  north  from  its  base  70  yards,  and  found  the 
elevation  of  the  top  to  be  25°  10V;  then,  measuring  due 
east  60  yards,  the  elevation  of  the  top  was  20°  4'.  What 
was  the  length  and  inclination  of  the  tree  ? 

Ans.  Length,  35.1  yards ;  inclination,  83°  11'. 

Ex.  19.  The  bearings  and  distances  being  as  in  Ex.  16, 
it  is  required  to  divide  the  tract  into  two  equal  parts  by  a 
line  running  from  the  first  corner.  The  bearing  of  the 
division  line  is  required. 

Ans.  1ST.  14°  59'  E.  27.66  chains  to  a  point  on  the  fifth 
side  1.61  from  beginning. 

Ex.  20.  The  boundaries  of  a  quadrilateral  are, — 1.  "N.  35J° 
E.  23  chains;  2.  K  75J0  E.  30.50  chains;  3.  S.  3J°  E.  46.49 
chains,  and  4.  "N.  66 J°  "W.  49.64  chains, — to  divide  the  tract 
into  four  equal  parts  by  two  straight  lines,  one  of  which 
shall  be  parallel  to  the  third  side.  Required  the  distance 
of  the  parallel  line  from  the  first  corner,  the  bearing  of  the 
other  division  line  and  its  distance  from  the  same  corner, 
measured  on  the  first  side. 

Ans.  Distance  of  parallel  division,  32.50  chains ;  bear- 
ing of  the  other,  S.  88°  22'  E. ;  distance  from  the  first 
corner,  5.99  chains. 


CHAPTER  IX. 

MERIDIANS,    LATITUDE,    AND    TIME. 


SECTION    I. 
MERIDIANS. 

409.  THE  meridian  of  a  place  is  a  true  north  and  south 
line  through  that  place ;  or  it  may  be  defined  to  he  a  great 
circle  of  the  earth  passing  through  the  pole  and  the  place. 

410.  As  it  is  of  great  importance  to  the  surveyor  to 
be  able  to  trace  accurately  a  meridian  line,  the  following 
methods  are  given.     Any  one  of  these  is  sufficiently  accu- 
rate for  his  purposes.     Those  which  require  the  employ- 
ment of  the  transit  or  the  theodolite  are  to  be  preferred, 
if  one  of  these  instruments  is  at  hand.     When  the  obser- 
vations are  performed  with  the  proper  care,  and  the  instru- 
ments are  to  be  depended  on,  the  line  may  be  run  within  a 
few  seconds  of  its  proper  position. 

411.  Although  the  methods  to  be  explained  in  the  follow- 
ing articles  are  in  theory  perfectly  accurate,  yet  the  results 
to  which  they  lead  cannot  be  relied  on  with  the  same  cer- 
tainty when   the   observations   are  made  with    surveyors' 
instruments,  as  if  the  larger  and  more  expensive  instru- 
ments to  be  found  in  fixed  observatories  were  employed. 
These  instruments  generally  rest  on  permanent  supports: 
their  positions  and  adjustments  may  therefore  be  tested,  and 
corrected  when  found  defective,  and  thus  their  proper  posi- 
tion be  finally  obtained  with  almost  perfect  accuracy.     Not 

307 


308 


MERIDIANS,  LATITUDE,  AND  TIME. 


[CHAP.  IX. 


so  with  the  theodolite  or  the  surveyors'  transit.  The  ad- 
justments in  their  position  must  be  made  at  the  time,  and 
renewed  for  every  fresh  observation.  The  results  alone  are 
to  be  corrected  by  subsequent  observation,  and  not  the 
position  of  the  instrument.  Notwithstanding  these  diffi- 
culties, which  must  always  prevent  his  attaining  the  pre- 
cision of  the  astronomer,  yet,  with  ordinary  care,  the  sur- 
veyor may  run  his  lines  with  all  the  accuracy  which  is 
necessary  for  his  operations. 

Problem  1. — To  run  a  meridian  line. 

412.  First  Method. — By  equal  altitudes  of  the  sun. 

Select  a  level    surface,   ex-  Fig.  199. 

posed  to  the  south,  and  erect 
an  upright  staff  upon  it. 
Around  the  foot  of  this  staff 
A  (Fig.  199)  as  a  centre  de- 
scribe a  circle.  Observe  care- 
fully the  point  B  at  which  the 
end  of  the  shadow  crosses  this 
circle  in  the  morning,  and 
likewise  the  point  C  where  it 
crosses  in  the  evening.  Bisect 
the  angle  BAG  by  the  line  NS, 
which  will  be  a  meridian.  If 
a  number  of  circles  be  de- 
scribed around  A,  several  observations  may  be  made  on  the 
same  day,  and  the  mean  of  the  whole  taken. 

If  the  staff  is  not  vertical,  let  fall  a  plumb-line  from  the 
summit,  and  describe  the  circles  around  the  point  in  which 
this  line  cuts  the  surface.  A  piece  of  tin,  with  a  small  cir- 
cular hole  through  it  for  the  sun's  rays  to  pass  through,  is 
better  than  the  top  of  the  staff,  the  image  being  definite. 

Where  much  accuracy  is  not  required,  the  above  method 
is  sufficient.  It  supposes  the  declination  of  the  sun  to  re- 
main unchanged  during  the  observation.  This  is  not  true 
except  at  the  solstices, — 21st  of  June  and  22d  of  December. 


SEC.  I.].  MERIDIANS.  309 

Those  days — or  at  least  a  time  not  very  rembte  from  them 
— should  therefore  be  chosen  for  determining  the  meridian 
by  this  method. 

413,  Second  Method.  —  By  a  meridian   observation  of  the 

North  Star. 


The  Pole  Star  (Polaris,  or  a  Ursce  Minoris)  is  situated 
very  nearly  at  the  North  Pole  of  the  heavens.  If  it  were 
exactly  so,  all  that  would  be  necessary  to  determine  the 
direction  of  the  meridian  would  be  to  sight  to  the  star  at 
anytime.  The  North  Star,  being,  however,  about  1J°  from 
the  pole,  is  only  on  the  meridian  twice  in  twenty-four 
hours. 

There  is  another  star,  (Alioth,)  in  the  tail  of  the  Great 
Bear,  ( Ursce  Majoris,}  which  is  on  the  meridian  very  neanly 
at  the  same  time  as  the  Pole  Star. 

The  constellation  in  which  Alioth  is  situated  is  one  of  the 
most  generally  known.  It  is  often  called  the  Plough,  the 
Dipper,  the  Wagon  and  Horses,  or  Charles's  Wain.  The 
two  stars  in  the  quadrangle  farthest  from  the  handle,  or 
tail,  are  called  the  Pointers,  from  the  fact  that  the  line 
joining  them  will,  when  produced,  pass  nearly  through  the 
Pole  Star.  The  star  in  the  handle  of  the  dipper,  nearest 
the  quadrangle,  is  Alioth. 

414.   To  determine  the  direction  of  the  meridian. 

Suspend  a  long  plumb-line  from  some  fixed  elevated 
point.  If  a  window  can  be  found  properly  situated,  a  staff 
may  be  projected  from  it  to  afford  a  support.  The  plum- 
met should  be  heavy,  and  be  allowed  to  swing  in  a  vessel 
of  water,  so  as  to  lessen  the  effect  of  the  currents  in  the 
air.  At  some  distance  to  the  south  of  the  line  set  two  posts, 
east  and  west  from  each  other,  making  their  tops  level,  and 
nail  upon  them  a  horizontal  board.  To  another  board 
screw  a  compass-sight.  This  may  be  moved  steadily 
to  the  east  or  west  upon  the  other  board.  Then,  some 
time  before  Polaris  is  on  the  meridian,  place  the  compass- 


310  MERIDIANS,   LATITUDE,  AND  TIME.  [CHAP.  IX. 

sight  so  that  by  looking  through  it  Alioth  may  be  hidden 
by  the  plumb-line.  As  the  star  recedes  from  the  line, 
move  the  sight,  so  as  to  keep  the  line  and  star  in  the  same 
direction ;  at  last  Polaris  will  also  be  covered  by  the  line. 
The  eye  and  plumb-line  are  then  very  nearly  in  the  me- 
ridian. If  the  time  is  noted,  and  Polaris  sighted  to  seven- 
teen minutes  after  the  former  observation,  the  meridian  will 
be  much  more  accurately  determined.  The  compass-sight 
may  now  be  firmly  clamped  till  morning.  In  making  the 
above  described  observation,  it  will  generally  be  necessary* 
for  an  assistant  to  illuminate  the  line  if  the  night  is  dark. 

415.   To  determine  the  time  Polaris  is  on  the  meridian. 

1.  Take  from  the  American  Almanac,  or  other  Ephemeris, 
the  sun's  right  ascension,  or  sidereal  time  of  mean  noon, 
for  the  noon  preceding  the  time  for  which  the  transit  is 
wanted.  The  sidereal  time  is  given  in  the  American  Al- 
manac for  mean  noon  at  Greenwich  (England)  for  every  day 
in  the  year,  and  may  be  calculated  for  any  other  meridian 
by  interpolation,  thus : — 

The  difference  between  the  sidereal  times  for  two  suc- 
cessive days  being  3  minutes  56.555  seconds,  say,  As  twenty- 
four  hours  is  to  the  longitude  expressed  in  time,  so  is  3  minutes 
56.555  seconds  to  the  correction  to  be  applied  to  the  sidereal  time 
at  noon  of  the  given  day  at  Greenwich.  This  correction — 
added  to  the  sidereal  time  taken  from  the  almanac  if  the 
longitude  be  west,  but  subtracted  if  it  be  east — will  give 
the  sidereal  time  at  mean  noon  at  the  given  place. 

The  above  correction,  having  been  once  determined  for 
the  given  place,  will  serve  for  all  the  calculations  that  may 
be  wanted. 

EXAMPLE. 

Ex.  1.  Let  it  be  required  to  find  the  sidereal  time  at 
mean  noon,  at  Philadelphia,  long.  5  h.  0  m.  40  sec.  W.,  on 
the  llth  of  August,  1855. 

The  sidereal  time  at  mean  noon,  Greenwich,  August  11, 


SEC.  I.].  MERIDIANS.  311 

is  9  hours,  17  minutes,  32.74  seconds,  as  taken  from  the 
American  Almanac  of  that  year. 

And,  As  24  h.  :  5  h.  0  m.  40.  s.  : :  3m.  56.555  s.  :  49.391. 

h.     m.        sec. 

Then,  sidereal  time  at' Greenwich,  mean  noon  9  17  32.74 
Correction  for  difference  of  long.  49.39 

Sidereal  time  at  Philadelphia,  mean  noon          9  18  22.13 

2.  Subtract  the  sidereal  time  above  determined  from  the 
right  ascension  of  the  star,  taken  from  the  same  almanac, 
increasing  the  latter  by  24  hours,  if  necessary  to  make  the 
subtraction  possible.  The  remainder  is  the  time  of  the 
transit  expressed  in  sidereal  hours. 

To  convert  these  into  solar  hours.  Say,  As  24  hours  is  to 
the  number  of  hours  in  the  above  time,  so  is  3  minutes  55.9 
seconds  to  the  correction.  This  correction,  subtracted  from 
the  sidereal  time,  will  give  the  mean  solar  time  of  the  upper 
transit. 

The  time  thus  determined  will  be  astronomical  time. 
The  astronomical  day  begins  at  noon,  the  hours  being 
counted  to  twenty-four.  The  first  twelve  hours,  therefore, 
correspond  with  the  hours  in  the  afternoon  of  the  same 
civil  day;  but  the  last  twelve  agree  with  the  hours  of  the 
morning  of  the  next  succeeding  day. 

Thus,  August  11,  8h.  15m.,  astronomical  time,  corresponds 
with    August  11,  8h.  15m.  P.M.,  civil  time; 
but      August  11, 16  h.  15  m.,  astronomical  time,  agrees  with 
August  12,  4  h.  15  m.  A.M.,  civil  time. 

If,  therefore,  the  number  of  hours  of  a  date  expressed  in 
astronomical  time  be  greater  than  twelve,  to  convert  it  into 
civil  time  the  days  must  be  increased  by  one  and  the  hours 
diminished  by  twelve. 

Required  the  time  of  the  upper  transit  of  Polaris,  Sep- 
tember 11,  1855,  for  Philadelphia. 


312  MERIDIANS,  LATITUDE,  AND  TIME.  [CHAP.  IX. 

Sidereal  time  at  mean  noon,  Greenwich, 

September  10 

Correction  for  Philadelphia 
Sidereal  time,  mean  noon,  at  Phila.  (A) 
Eight  ascension  of  Polaris,  Sept.  11  (B) 
(B)-(A) 

Correction  for  13  h.  50  m.  24  sec. 
Astronomical  time,  September  10  13  48 

agreeing  with  civil  time,  Sept.  11  1  48 


h.      m.         Bee. 

11  15  49.38 
49.39 

11  16  38.7T 

1    7    2.71 

13  50  23.94 

2  16.04 


7.90 
7.90  A.M. 


416.  The  times  of  the  upper  transit  of  Polaris  for  every 
tenth  day  of  the  year  is  given  in  the  following  table. 
The  calculation  is  made  for  the  meridian  of  Philadelphia, 
the  year  1855.  As  a  change  of  six  hours,  or  90°  of  longi- 
tude, will  only  make  a  change  of  one  minute  in  the  time 
of  the  transit,  the  table  is  sufficiently  accurate  for  any  place 
within  the  United  States  : — 

Time  of  Polaris  crossing  the  meridian,  upper  transit. 


Months. 

1st. 

nth. 

21st. 

Jan  uary  

h.    m. 

6  22  P.M. 

h.    m. 

5  43  P.M. 

h.       m. 

5.   3  P.M. 

February  

420   " 

340  " 

31" 

March  

229   « 

1  50  " 

1  11  " 

April  

0  27   " 

11  48A.M. 

11     9A.M. 

May  

10  30  A.M. 

9  50  " 

9  11  " 

June  

828   " 

749  " 

7  10  " 

July  

6  30   " 

5  51  « 

5  12  « 

August  

429   " 

3  50  " 

3  11  " 

September  

227   " 

148  " 

19" 

October  

0  30   " 

11  46  P.M. 

11     7  P.M. 

ISTo  vember  

10  24  P.M. 

9  44  " 

95" 

December... 

8  26   " 

746  " 

77" 

If  the  time  of  the  passage  of  the  star  for  any  day  not 
given  in  the  table  be  desired,  take  out  the  time  of  passage 
for  the  day  next  preceding,  and  deduct  four  minutes  for 


SEC.  I.]  MERIDIANS.  313 

each  day  that  elapses  between  the  date  in  the  table  and 
that  for  which  the  time  of  transit  is  required ;  or,  more  ac- 
curately, thus : — 

Say,  As  the  number  of  days  between  those  given  in  the  table  is 
to  the  number  between  the  preceding  date  and  that  for  which  the 
time  of  transit  is  desired,  so  is  the  difference  between  the  times  of1 
transit  given  in  the  table  to  the  time  to  be  subtracted  from  that 
corresponding  to  the  earlier  of  the  two  days. 

Let  the  time  of  transit,  August  27,  be  desired. 

Time. 

Aug.  21,  3  h.  11  m. 

Sept.    1, 

Difference 
As  lid.  :  6  d.  ::  44:  24; 

therefore  3  h.  11  m.  —  24  m.  =  2  h.  47  m.  is  the  time  re- 
quired. 

417.  If  the  time  of  the  lower  transit  be  desired,  it  may 
be  obtained  from  the  table  by  changing  A.M.  into  P.M.  and 
diminishing  the  minutes  by  2,  or  changing  P.M.  into  A.M.  and 
increasing  the  minutes  by  2. 

418.  The  above  table  is  calculated  for  the  year  1855.     It 
will,  however,  serve  for  the  observation  described  in  Art. 
414  for  many  years,  the  time  of  the    meridian  passage 
being  determined  in  that  method  by  the  time  of  Polaris 
and  Alioth  being  in  the  same  vertical.     When  the  time  is 
more  accurately  needed,  as  in  Method  3  (Art.  419)  for  deter- 
mining the  meridian,  it  will  be  necessary  to  correct  the 
numbers  in  the  table  for  the  years  that  elapse  between  1855 
and  the  current  year. 

The  Pole  Star  passes  the  meridian  about  21  seconds — 
more  accurately,  20.6  seconds  —  later  every  year  than  the 
preceding  one,  so  that  in  1860  the  time  will  be  1  minute, 
43  seconds  later  than  those  given  in  the  table ;  in  1870,  5 
minutes ;  in  1880,  8  minutes  35  seconds ;  and,  in  1890,  12 
minutes  later. 


314  MERIDIANS,  LATITUDE,  AND  TIME.  [CHAP.  IX. 

419.   Third  Method. — By  a  meridian  passage  observed  with  a 
transit  or  theodolite. 


Having  accurately  levelled  the  instrument,  sight  to  Po- 
laris when  on  the  meridian.  Then,  depressing  the  telescope, 
set  up  an  object  in  the  line  of  sight :  a  line  drawn  from  the 
instrument  to  that  object  will  be  a  meridian. 

In  observing  with  the  transit  or  theodolite  at  night,  it  is 
needful  that  the  wires  should  be  illuminated.  This  may  be 
done  by  an  assistant  reflecting  the  rays  of  a  lamp  into  the 
tube  by  a  sheet  of  white  paper. 

An  error  of  5  minutes  in  the  time  of  the  transit  of  Po- 
laris will  make  an  error  of  about  1J'  in  the  bearing  of  the 
star,  so  that  if  the  observation  is  not  made  near  the  proper 
time,  it  must  be  corrected. 

This  may  be  done  thus : — Deduct  the  star's  polar  distance 
from  the  complement  of  the  latitude.  Then  say,  As  sine 
of  this  difference  is  to  the  sine  of  the  polar  distance  of  the  star, 
(1°  28'  at  present,)  so  is  sine  of  the  error  in  time  (expressed  in 
degrees)  to  the  sine  of  the  bearing  of  the  star.  East  if  the  time 
be  too  early,  but  west  if  it  be  too  late. 

The  time  is  reduced  to  degrees  by  multiplying  by  15: 
thus,  5  minutes  =  1°  15'. 

EXAMPLE. 

Required  the  bearing  of  Polaris  5  minutes  after  the  upper 
meridian  passage,  the  latitude  of  the  place  being  40°. 

50°  -  1°  28'  =  48°  32' 

As  sine  of  48°  32'     Ar.  Co.  0.125320 

:    sine  of  star's  polar  distance       1°  28'  8.408161 

: :  sine  of  time,  in  degrees,  1°  15'  8.338753 

:    sine  of  star's  bearing  V  37"  W.  6.872234 

420.  Fourth  Method. — By  an  observation  of  Polaris  at  its 
greatest  elongation. 

As  a  circumpolar  star  revolves  round  the  pole,  it  gradu- 
ally recedes  from  the  meridian  towards  the  west  until  it 


SEC.  L]  MERIDIANS.  315 

attains  its  most  remote  point:  here  it  apparently  remains 
stationary,  or  at  least  appears  to  move  directly  towards  the 
horizon  for  a  few  minutes,  and  then  gradually  moves  east- 
ward towards  the  meridian,  which  it  crosses  below  the  pole. 
Continuing  \  its  course,  in  about  six  hours  it  reaches  its 
greatest  eastern  deviation,  when  it  again  becomes  sta- 
tionary. When  most  remote  from  the  meridian,  it  is  said 
to  have  its  greatest  elongation. 

As  the  star  is  apparently  stationary  at  the  time  of  its 
greatest  eastern  or  western  elongation,  this  time  is  a  very 
favorable  one  for  observing  it.  A  variation  of  a  few 
minutes  in  the  time  will  then  make  no  appreciable  error 
in  the  bearing  of  the  line. 

421.  The  subjoined  table  contains  the  times  of  the  great- 
est eastern  or  western  elongations,  according  as  the  one  or 
the  other  occurs  at  a  time  of  day  favorable  for  observa- 
tion. The  times  of  greatest  elongations  are  calculated 
thus :  Take  from  one  of  the  almanacs  mentioned  in  Art.  415 
the  polar  distance  of  the  star  at  the  given  time,  and  call  it 
P.  Call  the  latitude  of  the  place  L.  Then  find  the  semi- 
diurnal arc  by  the  following  formula: — 

E  .  cosine  x  =  tan.  P  .  tan.  L. 

Reduce  x  to  time  by  dividing  by  15,  calling  the  degrees 
hours,  and  correct  for  the  sidereal  acceleration :  the  result 
will  be  the  semidiurnal  arc  expressed  in  time.  Call  it  L 
Then,  if  T  be  the  time  of  greatest  elongation,  and  T'  be 
the  time  of  the  upper  meridian  passage  of  the  star,  T  =  T' 
+  t  or  T'  —  £,  according  as  the  time  of  the  western  or 
eastern  elongation  is  desired. 

The  hour  angle  for  Polaris  at  its  greatest  elongation, 
July  1,  1855,  in  lat.  40°  IS".,  was  5  hours  54  minutes ;  but, 
as  the  polar  distance  of  the  star  is  diminishing  at  the  rate 
of  19.23"  per  annum,  the  semidiurnal  arc  is  slowly  in- 
creasing. The  change  is  so  small,  however, — being  about 
one  second  per  year, — that  it  may  be  entirely  neglected. 
As  the  time  of  the  meridian  passage  of  the  star  is  later  by 
20.6  seconds  each  year  than  the  preceding  one,  the  times 


316 


MERIDIANS,  LATITUDE,   AND  TIME. 


[CHAP.  IX. 


of  greatest  eastern  and  greatest  western  elongation  will  be 
similarly  affected :  in  1860  they  will  be  1  minute  43  seconds 
later  than  the  times  given  in  the  table ;  in  1870,  5  minutes ; 
and,  in  1880,  8  minutes  35  seconds  later. 

422.   Table  of  Times  of  Greatest  Elongation  of  Polaris  for 
1855.     Latitude,  40°  K 


Months. 

1st. 

nth. 

21st. 

January... 
February.. 
March  
April  
Mav. 

West 
West 
West 
East 
East 

h.     m. 
0  16  A.M. 

10  14  P.M. 
8  23    " 
6  33  A.M. 
4  35    " 

h.     m. 

11  37  P.M. 
9  35    « 
744    « 
5  54  A.M. 
3  56    " 

h.     m. 

10  57  P.M. 
855    " 
74" 

5  15  A.M. 

3  17    " 

June  

East 

2  34    " 

1  55    " 

1  15    " 

July  

East 

0  36    " 

11  53  P.M. 

11  14  P  M. 

August  .... 
September 
October  ... 
November 
December 

East 
East 
West 
West 
West 

10  31  P.M. 
829    « 
6  24  A.M. 
4  22    " 
2  24    " 

951    " 
750    " 
5  44  A.M. 
342    " 
145    « 

9  12    « 
711    " 

5     5  A.M. 

33" 
1    5    " 

The  above  table  is  calculated  for  lat.  40°,  for  which  lati- 
tude the  hour  angle  is  5  h.  54  m.    6  sec. ; 
for  latitude  50°  the  hour  angle  is          5      52         2, 
and  for  lat.  30°    "            "         "          5      55        38 ; 
therefore,  for  lat.  50°  the  eastern  elongation  occurs   two 
minutes  later,  and  the  western  two  minutes  earlier,  than 
those  given  in  the  table;    for  lat.  30°  the  times  of   the 
eastern  elongation  must  be  diminished,  and  those  of  the 
western  increased,  by  1  minute  32  seconds. 

423.  The  observation  for  the  meridian  is  made  as  directed 
Art.  414.  Suspend  the  plumb-line,  and,  having  placed  the 
compass-sight  on  the  table,  as  the  star  moves  one  way  move 
the  sight  the  other,  so  as  to  keep  the  star  always  hid  by 
the  line.  At  the  time  of  greatest  elongation  the  star  will 
appear  stationary  behind  the  line.  Clamp  the  board  to 
which  the  compass-sight  is  attached.  If  the  plumb-line  is 
suspended  from  a  point  that  is  not  liable  to  derangement, 


SEC.  L] 


MERIDIANS. 


317 


the  remainder  of  the  work  may  be  left  till  daylight ;  other- 
wise, let  an  assistant  take  a  short  stake,  with  a  candle 
attached  to  it,  to  a  distance  of  8  or  10  chains.  He  may 
then  be  placed  exactly  in  line  with  the  plumb.  "When  the 
stake  has  been  so  adjusted,  it  should  be  driven  firmly  into 
the  ground  and  its  position  again  tested. 

Measure  accurately  the  distance  between  the  compass- 
sight  and  the  stake.  Call  it  D.  Take  the  azimuth  of  the 
star  from  the  following  table  and  call  it  A. 

D . tan.  A 
Calculate  x  — — , 

xt 

and  set  off  the  distance  x  to  the  east  or  west  of  the  stake, 
according  as  the  western  or  eastern  elongation  was  observed. 
The  point  thus  determined  will  be  on  the  meridian  passing 
through  the  compass-sight.  Permanent  marks  may  then 
be  fixed  at  any  convenient  points  in  this  line. 

If  a  transit  or  theodolite  is  at  hand,  direct  the  telescope 
to  the  stake  first  set  up.  Turn  it  through  an  angle  equal 
to  the  azimuth :  it  will  then  be  in  the  meridian :  or  direct 
the  telescope  to  the  star  when  at  its  greatest  elongation, 
and  then  turn  the  plate  through  an  angle  equal  to  the 
azimuth. 


424.  The  azimuth  of  a  star  is  its  bearing,  and  may  be 
determined  by  the  following  formula, — A  being  the  azi- 
muth, L  the  latitude  of  the  place,  and  P  the  polar  distance 
of  the  star: — 

Q.       .        K  .  sin.  P 

Sm.  A  = — . 

cos.  L 

Azimuths  of  the  Pole  Star  at  its  Greatest  Elongation. 


Lat. 

1855. 

I860. 

1865. 

1870. 

o 

30 
35 
40 
45 

Of            II 

1  41  21 
1  47  11 
1  54  37 
2    4  11 

0          1             II 

1  39  32 
1  45  14 
1  52  32 
2     1  55 

1  37  42 
1  43  16 
1  50  27 
1  59  35 

1  35  49 
1  41  19 

1  48  20 
1  57  18 

318  MERIDIANS,  LATITUDE,  AND  TIME.  [CHAP.  IX. 

The  above  are  calculated  from  the  mean  place  of  the  star 
as  given  in  Loomis's  "Practical  Astronomy." 

425.  Fifth  Method. — By  equal  altitudes  of  a  star. 

If  a  theodolite  or  a  transit  with  a  vertical  arc  is  at  hand, 
the  meridian  may  be  run  very  accurately  by  observing  a 
star  when  at  equal  altitudes  before  and  after  passing  the 
meridian. 

For  this  purpose  select  a  star  situated  near  the  equator, 
and,  having  levelled  the  instrument  with  great  care,  take 
the  altitude  of  the  star  about  two  or  three  hours  before  it 
passes  the  meridian,  and  notice  carefully  the  horizontal 
reading.  "When  the  star  is  about  as  far  to  the  west  of  the 
meridian,  set  the  telescope  to  the  same  elevation,  and  fol- 
low the  star  by  the  horizontal  motion  until  its  altitude  is 
the  same  as  before,  and  again  notice  the  reading. 

Then  if  the  zero  is  not  between  the  two  observed  read- 
ings, take  half  their  sum,  and  turn  the  telescope  until  the 
vernier  is  at  that  number  of  degrees  and  minutes :  the  tele- 
scope will  then  be  in  the  meridian.  If  the  veinier  has 
passed  the  zero,  add  360  to  the  less  reading  before  taking 
the  sum. 

Thus,  if  the  first  reading  were  150°  37'  30",  and  the 

431°  2'  30" 
second  280°  25',  the  half  sum  —  —  =  215°  31'  15" 

tU 

would  be  the  reading  for  the  meridian. 

Instead  of  taking  the  readings,  a  stake  may  be  set  up  at 
any  distance — say  ten  chains — in  each  observed  course :  then 
bisect  the  line  joining  the  stakes,  and  run  a  line  from  the 
instrument  to  the  point  of  bisection. 

The  mean  of  a  few  observations  taken  in  this  manner 
will  determine  the  meridian  with  considerable  precision. 


SEC.  II,] 


LATITUDE. 


319 


SECTION  II. 

LATITUDE, 

THE  latitude  of  a  place  may  be  determined  in  various 
modes. 

426.  First  Method. — By  a  meridian  altitude  of  the  Pole  Star. 

The  altitude  of  the  pole  is  equal  to  the  latitude  of  the 
place.  Take  the  altitude  of  Polaris  when  on  the  meridian, 
and  from  the  result  subtract  the  refraction  taken  from  the 
following  table.  Increase  or  diminish  the  remainder  by  the 
polar  distance  of  the  star  according  as  the  lower  or  upper 
transit  was  observed :  the  result  will  be  the  latitude. 

427.  Eefraction  to  be  taken  from  the  apparent  latitude. 


App. 

Alt. 

Eef. 

App. 
Alt. 

Kef. 

App. 
Alt. 

Eef. 

App. 

Alt. 

Kef. 

App. 

Alt. 

Kef. 

0 

20 

/  // 
2  39 

0 

30 

1  40 

o 

40 

/   // 

1  9 

o 

50 

/   // 

0  49 

0 

60 

0  34 

21 

2  30 

31 

1  37 

41 

1  7 

51 

0  47 

61 

0  32 

22 

2  23 

32 

1  33 

42 

1  5 

52 

0  45 

62 

0  31 

23 

2  16 

33 

1  29 

43 

1  2 

53 

0  44 

63 

0  30 

24 

2  10 

34 

1  26 

44 

1  0 

54 

0  42 

64 

0  28 

25 

2  4!|  35 

1  23 

45 

0  58 

55 

0  41 

65 

0  27 

26 

1  59 

36 

1  20 

46 

0  56 

56 

0  39 

66 

0  26 

27 

1  54 

37 

1  17 

47 

0  54 

57 

0  38 

67 

0  25 

28 

1  49 

|38 

1  14 

48 

0  52 

58 

0  36 

68 

0  24 

29 

1  45! 

39 

1  12  49 

0  50 

59 

0  35 

69 

0  22 

428.  Second  Method. — Take  the  altitude  of  the  star  six 
hours  before  or  after  its  meridian  passage.     The  result, 
corrected  for  refraction,  will  be  the  latitude. 

429.  Third  Method. — By  a  meridian  altitude  of  the  sun. 


Take  the  meridian  altitude  of  the  upper  or  the  lower 
limb  of  the  sun,  and  correct  for  refraction.     The  result, 


320  MERIDIANS,  LATITUDE,  AND  TIME.  [CHAP.  IX. 

increased  or  diminished  by  the  semidiameter  of  the  sun 
according  as  the  lower  or  the  upper  limb  was  observed,  will 
be  the  altitude  of  the  sun's  centre.  (The  apparent  semi- 
diameter  of  the  sun  is  given  in  the  American  Almanac  for 
every  day  of  the  year.) 

To  the  altitude  of  the  sun's  centre,  add  his  declination 
(taken  from  the  same  almanac)  if  south,  but  subtract  it  if 
north :  the  result  subtracted  from  90°  will  give  the  latitude. 

Instead  of  the  sun,  a  bright  star,  the  declination  of  which 
is  small,  may  be  observed. 

430.  If  the  exact  direction  of  the  meridian  is  not  known, 
the  telescope  must  be  fixed  on  the  body  some  time  before  it 
is  south.     As  the  sun  or  star  approaches  the  meridian  its 
altitude  increases,  and  it  will  therefore  rise  above  the  hori- 
zontal wire.     Move  the  telescope  in  altitude  and  azimuth 
so  as  to  follow  the  body  until  it  ceases  to  leave  the  wire. 
The  reading  will  then  give  the  observed  meridian  altitude. 
The  altitude  alters  very  slowly  for  some  minutes  before  and 
after  its  meridian  passage,  thus  affording  ample  time  to 
direct  the  telescope  accurately  towards  the  object. 

431.  Fourth  Method. — By  an  observation  of  a  star  in  the 
prime  vertical. 

Any  great  circle  passing  through  the  zenith  is  called  a 
vertical  circle.  All  such  circles  are  perpendicular  to  the 
horizon. 

That  vertical  circle  which  is  perpendicular  to  the  meridian 
is  called  the  prime  vertical :  it  cuts  the  horizon  in  the  east 
and  west  points. 

Level  the  plates  of  the  transit  or  theodolite  carefully,  and 
direct  the  telescope  to  the  east  or  west,  so  that  it  may  move 
in  the  prime  vertical  or  nearly  so.  Then,  having  selected 
some  bright  star  which  passes  the  meridian  a  little  south  of 
the  zenith,  (the  declination  of  such  a  star  is  rather  less  than 
the  latitude  of  the  place,)  observe  the  time  of  its  crossing 
the  vertical  wire  of  the  telescope  before  passing  the  meridian, 
and  again,  when  in  the  west,  after  its  meridian  passage.  Let 


SSLCJ.  II.]  LATITUDE.  321 

these  times  be  called  T  and  T'.  Let  the  interval  "between 
T  and  T'  be  called  x,  which  must  be  reduced  to  sidereal 
time  by  adding  to  the  solar  time  3  minutes  56.55  seconds 
for  24  hours,  or  9.85  seconds  per  hour ;  also,  let  L  be  the 
latitude  of  the  place,  and  1)  be  the  declination  of  the  star. 

E.  tan.  D 

Then  tan.  L  = 

cos.  Jz 

Thus,  for  example,  the  transit  of  a  Lyrce  over  the  prime 
vertical  was  observed  July  1,  1855,  at  10  h.  43  m.  4  sec., 
and  again  at  13  h.  3  m.  48  sec.,  mean  solar  time.  Ke- 
quired  the  latitude, — the  apparent  right  ascension  of  the 
star  (as  given  in  the  American  Almanac)  being  18  h.  32  m. 
4  sec.,  and  the  declination  38°  39'  0.4". 

Here  the  interval  is  2  h.  20  m.  44  sec.,  solar  time. 
Eeduction  23 


2)2   21    7 

sec.  =  17°  38'  22". 
A.  C.  0.020915 
9.902940 
9.923855 

1  h.  10  m.  33.5 
17°  38'  22" 
38°  39'  0.4" 
40°  0'4" 

Cos.  \x 
tan.  D 
tan.  L 

432.  Half  the  sum  of  the  observed  times  is  the  time  of 
meridian  passage  in  mean  solar  time.  If  this  is  reduced  to 
sidereal  time  and  increased  by  the  sidereal  time  of  mean 
noon  at  the  given  place,  the  result  should  be  equal  to  the 
right  ascension  of  the  star. 

In  the  example  before  us  the  times  of  observation  are 

h.        m.        sec. 

10  43  4 

and                                                                   13  3  48 

Sum                                                               2)23  46  52 

Half  sum                                                         11  53  26 

Reduction  for  sidereal  time  1  57 

(A)                                            11  55  23 

21 


322  MERIDIANS,  LATITUDE,  AND  TIME.  [CHAP.  IX. 

Sidereal  time,  mean  noon,  at  Greenwich  6  h.  35  m.  54  sec. 

Add  for  difference  of  meridians  49 

~6 36 33 


Add  (A)  18       31         56 

Eight  ascension  of  star  18       32  4 

Error  in  position  of  the  instrument  8" 

A  slight  error  in  the  position  of  the  instrument  will  make 
no  appreciable  error  in  the  result.  Hence,  this  method 
affords  perhaps  the  best  means  of  determining  the  latitude. 


SECTION  III. 

TO  FIND  THE  TIME  OP  DAY, 

433.  First  Method, — IF  a  good  meridian  line  has  been 
run,  the  transit  or  theodolite  may  be  placed  in  that  line, 
and,  being  well  levelled,  the  telescope,  if  adjusted  by  being 
directed  to  the  meridian  mark,  will,  when  elevated,  move  in 
the  meridian. 

Observe  the  time  that  the  western  limb  of  the  sun  comes 
to  the  vertical  wire,  and  also  when  the  eastern  limb  leaves 
it.  The  mean  between  these  will  be  the  time  that  the  centre 
of  the  sun  is  on  the  meridian,  or  apparent  noon.  Increase  or 
diminish  the  observed  time  of  the  passage  of  the  centre 
by  the  equation  of  time  according  as  the  sun  is  too  slow  or 
too  fast,  and  the  result  will  be  the  time  of  mean  noon  as 
given  by  the  watch.  The  difference  between  this  and  twelve 
hours  will  be  the  error  of  the  watch. 

434.  Second  Method. — Calculate  the  time  that  a  fixed  star 
having  but  little   declination  will  pass   the  meridian   as 
directed  for  Polaris,  Art.  415.     Then  the  difference  between 
the  observed  and  the  calculated  time  will  be  the  error  of 
the  watch. 


SEC.  III.]  TO  FIND  THE  TIME   OF  DAY.  323 

435.  Third  Method. — If  the  meridian  line  has  not  been 
determined,  the  time  may  be  obtained  by  an  altitude  of  the 
sun  or  of  a  star  when  out  of  the  meridian. 

Take  the  altitude  of  the  sun  when  three  or  four  hours 
from  the  meridian,  noting  the  time  by  the  watch,  and  correct 
it  for  refraction  and  semidiameter.  The  altitude  of  the 
upper  limb  should  be  taken  in  the  afternoon,  and  the 
lower  in  the  morning,  as  the  wire  then  crosses  the  face  of 
the  sun  before  the  observation,  and  may  be  distinctly  seen. 

Call  the  altitude  of  the  sun  A,  the  polar  distance  D,  the 
latitude  L,  and  the  hour  angle  H. 

Then  sir,*  J  H  =  CM.  »  (A  +  L  +  D)  «n.  »  (L  +  D  -  A), 

sin.  D  .  cos.  L 

or,  if  S  =  J  (A  +  L  +  D),  then  S  —  A  =  }  (L  +  D  —  A), 

and  sin.*  J  H  =  00*8  .Bin.  (S  -  A^ 

sin.  D  .  cos.  L 

RULE. 

Call  the  corrected  altitude  A.  From  the  Ephemeris  take 
the  sun's  declination  at  the  time  of  observation,  (the  watch- 
time  will  be  sufficiently  accurate) ;  if  north,  subtract  it  from 
90°,  but  if  south,  add  it  to  90°  :  the  result  will  be  the  sun's 
polar  distance,  which  call  D.  Call  the  latitude  of  the  place 
L.  Let  S  =  \  (A  -f  L  +  D).  Add  together  Ar.  Co.  sin.  D, 
Ar.  Co.  cos.  L,  cos.  S,  and  sin.  (S  —  A),  divide  the  result 
by  2,  and  the  quotient  will  be  the  sine  of  half  the  hour 
angle  of  the  sun  at  the  time  of  observation.  If  the  obser- 
vation is  made  in  the  afternoon,  the  hour  angle  reduced  to 
time  is  the  apparent  time ;  but,  if  the  observation  is  in  the 
morning,  the  hour  angle  subtracted  from  12  is  the  apparent 
time.  To  the  apparent  time  apply  the  equation  of  time, 
and  the  result  is  the  mean  time  of  the  observation.  The 
difference  between  the  calculated  time  and  that  shown  by 
the  watch  is  the  error  of  the  watch. 

Several  observations  may  be  made  in  the  course  of  a  few 
minutes,  and  the  mean  of  the  results  taken.  If  the  obser- 
vation is  carefully  made  with  a  good  transit  or  theodolite, 


324  MERIDIANS,  LATITUDE,  AND  TIME.  [CHAP.  IX. 

the  time  obtained  by  this  method  will  not  differ  more  than 
a  small  fraction  of  a  minute  from  the  true  time. 

436.  If  a  star  is  observed  instead  of  the  sun,  the  mode 
of  calculation  is  the  same.     The  hour  angle  will  then  be  in 
sidereal  hours,  which  must  be  converted  into  solar  hours. 
The  result,  added  to  or  subtracted  from  the  time  of  the 
meridian  passage  of  the  star,  according  as  the  observation 
was  made  after  or  before  the  star  had  passed  the  meridian, 
will  give  the  mean  time  of  observation. 

437.  If  two  altitudes  of  the  sun  or  a  star  be  taken, -and 
the  times  noted  by  a  watch,  the  true  time  and  the  latitude 
may  both  be  found.     But,  as  other  and  preferable  methods 
have  already  been  given  for  finding  the  latitude,  it  is  un- 
necessary to  give  the  rule  here. 


CHAPTER  X. 

VARIATION  OF  THE  COMPASS.* 

438,  IT  has  been  mentioned  (Art.  268)  that  the  magnetic 
and  the  geographical  meridian  do  not  generally  coincide ; 
the   difference  between   the   directions  of ,  the   two  being 
called  the  variation  of  the  compass.     If  this  variation  were 
constant,  it  would  be  of  no  practical  importance  to  the  sur- 
veyor.    A  line  run  by  the  compass  at  one  time  could  be 
retraced  on  the  same  bearing  at  any  other.     The  variation 
is,  however,  subject  to  continual  changes, — some  of  them 
having  a  period  of  many  years,  perhaps  several  centuries, 
others  being  annual  or  diurnal,  and  some  accidental  or  tem- 
porary. 

439.  Secular  Change.    From  the  time  of  the  earliest 
observations  made  in  this  country  on  the  position  of  the 
magnetic  needle  till  about  the  commencement  of  the  pre- 
sent century,  the  north  point  was  gradually  moving  to  the 
west.     Since  then,  the  direction  of  its  motion  has  been  re- 
versed.    This  motion  constitutes  what  is  called  the  secular 
change.     To  give  an  idea  of  the  extent  of  this  deviation, 
the  following  table  of  observations,  made  at  Paris,  is  pre- 
sented : — 


Year. 

1541. 

7° 

Variation. 

East. 
30'     " 

u 
it 

10  West. 

55       " 
5       " 
34       " 

Year. 

1816, 
1823. 

22 
22 

Variation. 

0  25'  West. 
23 
20       " 
5       " 
12       « 
3       « 
17       " 

1580 

11 

1618. 
1663. 
1700. 

8 
0 
8 

1827 

..  .22 

1828. 
1829. 
1835. 

22 
22 
22 

1780. 

19 

1805. 

22 

1853. 

20 

1814. 

22 

*  This   subject,  in  its  connections  with   Land   Surveying,   -was  first  fully 
developed  by  Professor  Gillespie,  in  his  Treatise  before  referred  to. 

325 


326 


VARIATION  OF  THE  COMPASS. 


[CHAP.  X. 


From  this  table,  it  appears  that  in  1580  the  needle  had 
attained  its  greatest  eastern  deviation.  From  that  time  to 
about  the  year  1814  it  moved  towards  the  west,  the  great- 
est deviation  being  22°  34'.  Since  1814  it  has  been  moving 
to  the  east. 

From  observations  made  at  various  places  in  Europe  and 
America,  it  appears  that  similar  changes  have  been  going 
on  throughout  all  these  countries. 

440.  The  following  table,  mostly  taken  from  the  "  Report 
of  the  Superintendent  of  the  United  States'  Coast  Survey" 
for  1855,  gives  the  variation  and  secular  change  for  some 
of  the  more  important  places  in  this  country : — 


Locality. 

Lat. 

Lou. 

Date. 

Variation. 

Change 
in  1850. 

Montreal,  C.W.  .. 

45°  30' 

73°  35' 

1850 

-f-   9°  28' 

-f4' 

Toronto,      "  .... 

49°  40' 

79°  21' 

1850 

1°  36' 

Burlington,  Vt.  .. 

44°  27' 

73°  10' 

1855 

9°  57'.  1 

4'.  9 

Portland,  Me  

43°  39' 

70°  16' 

1851 

11°  41' 

Boston,  Mass  

44°  20' 

71°    2' 

1854 

9°  31' 

5'.  2 

Providence,  R.I  .. 

41°  50' 

71°  24' 

1855 

9°  31'.5 

6'.0 

New  Haven,  Conn 

41°  18' 

72°  54' 

1845 

6°  17'.3 

4'.  8 

New  York  City.... 

40°  43' 

74°    0' 

1845 

6°  25'.3 

5'.2 

Albany   N  Y   

42°  39' 

73°  44' 

1836 

6°  47' 

7'.2 

Philadelphia,  Pa.. 

39°  58' 

75°-  10' 

1855 

4°  31'.7 

6/8 

Pittsburg,  Pa  

40°  26' 

79°  58' 

1845 

33' 

3'.  5 

Wilmington,  Del.. 

39°  45' 

75°  34' 

1846 

2°  30'.  7 

Baltimore,  Md.  ... 

39°  16' 

76°  34' 

1847 

2°  18'.  6 

Washington,  D.C.. 

38°  53' 

77°    1' 

1855 

2°  25' 

5'.0 

Petersburg,  Va  .. 

37°  14' 

77°  24' 

1852 

0°  26'.5 

Columbia,  S.C.    .. 

34° 

81°    2' 

1854 

—  3°    1'.7 

Savannah,  Ga.    .. 

32°    5' 

81°    5' 

1852 

—  3°40'.3 

Cincinnati,  0..    .. 

39°    6' 

84°  22' 

1845 

—  4°    4' 

4' 

Richmond,  Ind    .. 

39°  49' 

84°  47' 

1845 

—  4°  52' 

4' 

Detroit,  Mich..    .. 

42°  24' 

82°  58' 

1840 

—  2°    0' 

1' 

San  Francisco,  Cal. 

37°  48' 

122°  27' 

1852 

—  15°  27' 

The  above  are  derived  from  the  best  data  that  could  be 
procured ;  but  many  of  the  observations  are  doubtless  very 
imperfect. 

441,  Line  of  no  Variation.  From  a  map  published  by 
Professor  Loomis,  it  appears  that  in  1840  the  lines  of  equal 
variation  crossed  the  United  States  in  a  direction  to  the  east 
of  south,  tending  more  to  the  east  in  the  New  England 
States.  At  that  date,  the  line  of  no  variation  passed  a  little 


VARIATION  OF  THE  COMPASS.  327 

to  the  west  of  Pittsburg  and  to  the  east  of  Kaleigh,  N.C., — 
all  those  portions  of  the  country  to  the  east  of  that  line 
having  western  variation.  From  a  similar  map,  published 
in  the  Report  above  referred  to,  it  appears  that  the  line 
of  no  variation  had  shifted  to  the  west  a  few  miles  since 
that  time.  It  also  results  from  the  calculations  in  the  same 
report,  that  the  rate  of  change  in  variation  has  now  attained 
its  maximum,  and  is  beginning  to  diminish. 

442.  As  it  is  frequently  of  importance  to  know  the 
former  variation,  the  following  information  is  added : — 

The  variation  in 

Burlington,  Vt,  in  1792  7°  38'  W.;  1818,  7°  30'  W. 

Salem,  Mass.,  1781  7°    2'  W.;  1805,  5°  57'  W. 

New  Haven,  Ct,      1761      .  5°  47'  W. ;  1775,  5°  25'  W. 

"        "  1819  4°  35'  W. 

New  York,  1686  8°  45'  W. ;  1750,  6°  22'  W 

"       "  1789  4°  20'  W. ;  1824, 4°  40'  W 

Philadelphia,  1710  8°  30'  W. ;  1750,  5°  45'  W 

1793  1°30'W.;  1837,  3°  52' W 

443.  From  the  table,  (Art.  440,)  the  variation  for  any 
time  not  far  remote  from  those  given  may  readily  be  found. 
This  will  also  apply  for  places  not  very  far  distant  from  the 
line  of  equal  variation  passing  through  that  place.     As, 
however,  the  rate  of  change  varies,  calculations  based  on 
such  a  table   can   only  be   considered   correct  when   the 
interval  of  time  is  comparatively  small.     In  all  cases,  when 
it  can  be  done,  the  variation  should  be  found  by  direct  ob- 
servation by  the  methods  explained  in  the  next  article. 

V444.   To  determine  the  change  in  variation  by  old  lines. 

As  the  rate  of  change  varies,  the  above  rule  can  only  be 
considered  as  true  when  the  interval  of  time  has  not  been 
great.  If  a  number  of  years  have  elapsed  since  the  prior 
survey,  and  no  observations  can  be  found  relating  to  the 
immediate  neighborhood,  the  change  of  variation  can  be 


328  VARIATION  OF  THE  COMPASS.  [CHAP.  X. 

found,  nearly,  by  comparison  with  other  places  where  such 
observations  have  been  made. 

"When  any  well-established  marks  can  be  found,  the 
change  may  be  determined  by  taking  the  bearings  of  these 
and  comparing  them  with  the  records.  The  difference  will 
give  the  change  that  has  taken  place  between  the  dates  of 
the  two  surveys. 

If  the  two  marks  are  not  on  the  same  line,  they  may  still 
be  used  for  this  purpose.  Thus,  according  to  an  old  deed, 
the  bearings  of  three  adjacent  sides  of  a  tract  were  as 
follows, — viz. :  1.  Beginning  at  a  marked  locust,  N".  60J°  E. 
200  perches  to  a  chestnut ;  2.  1ST.  25  J°  E.  183  perches  to  a 
post;  3.  K  45°  E.  105.3  perches  to  a  white-oak.  The 
locust  is  gone,  but  the  stump  remains,  and  the  white-oak  is 
still  standing.  The  intermediate  corners  are  entirely  lost. 

Setting  the  instrument  over  the  stump,  run  jN".  60 !°  E.  200 
perches ;  thence  K".  25  J°  E.  183  perches ;  and  thence  K  45° 
E.  105.3  perches. 

If  no  change  had  taken  place  in  the  Fig.  200. 

variation,  and  both   surveys  had  been  ^^ 

accurately  made,  the  last  distance 
would  have  been  terminated  at  the 
white-oak.  Instead  of  this,  however, 
the  tree  bears  S.  54°  25'  E.  2.93  perches. 
Fig.  200  is  a  draft  of  the  above. 

From  the  bearings  of  AB,  BC,  and 
CD,  calculate  that  of  AD,  which  (Art. 
350)  will  be  found  to  be  K  43°  59'  E. 
470.38  perches.  This,  therefore,  was 
the  bearing  and  distance  of  AD  at  the 
time  of  the  former  survey.  It  is  now  the  bearing  and  dis- 
tance of  AD'. 

With  the  latitude  and  departure  of  AD'  and  that  of  DD', 
calculate  the  present  bearing  and  distance  of  AD  (Art.  350.) 
It  will  be  found  to  be  1ST.  47°  54'  E.  476.25  perches.  The 
change  of  variation  has  therefore  been  3°  55'  W.  There  is 
likewise  a  variation  of  5.87  perches  in  the  measurement, 
from  which  it  is  inferred  that  the  chain  used  in  the  former 
survey  was  101.25  links  in  length,  or  1J  links  too  long. 


VARIATION  OF   THE   COMPASS.  329 

In  order,  therefore,  correctly  to  trace  the  lines  of  the  tract, 
the  vernier  of  the  compass  must  be  set  3°  55'  W.,  and  all 
the  distances  be  increased  1J  links  per  chain,  or  1J  perches 
per  hundred.  The  magnetic  bearings  and  the  distances  of 
the  three  sides  are  now,— 1.  N.  64°  25'  E.  202.5  perches; 
2.  K  29°  10'  E.  185.3  perches;  3.  K  48°  55'  E.  106.6 
perches. 

445.  Diurnal  Change.  If  the  position  of  the  needle  be 
accurately  noted  at  sunrise  on  a  clear  summer  day,  and  the 
observation  be  repeated  at  intervals,  it  will  be  found  that 
the  north  pole  will  gradually  be  deflected  to  the  west,  attain- 
ing its  maximum  deviation  about  2  or  3  o'clock.     During 
the  afternoon  it  will  gradually  return  towards  its  former 
position,  which  it  will  regain  about  8  or  9  o'clock  in  the 
evening.      This    deviation    from   the    normal   position   is 
known  as  the  diurnal  change.     It  amounts  sometimes  to  as 
much  as  a  quarter  of  a  degree,  being  greater  in  a  clear  day 
than  when  the  sky  is  overcast,  and  not  being  perceptible 
if  the  day  is  entirely  cloudy.     It  is  likewise   greater  in 
summer  than  in  winter. 

In  consequence  of  this  diurnal  change,  it  is  evident  that 
a  line  run  in  the  morning  cannot  be  retraced  with  the 
same  bearings  at  noon.  The  surveyor  should  therefore 
record  not  merely  the  date  at  which  a  survey  is  made,  but 
also  the  time  of  day  at  which  any  important  line  was  run, 
and  also  the  state  of  the  weather,  whether  clear  or  other- 
wise. 

446.  Irregular  Changes.      Besides    the   seculai    and 
diurnal  changes,  the  needle  is  subject  to  disturbance  from 
the  passage  of  thunder  storms,  or  from  the  occurrence  of 
aurora  boreali.     It  is  likewise  sometimes  violently  agitated 
when  no   apparent  cause  exists.      Such  disturbances  pro- 
bably result  from   the   occurrence  of  a  distant  magnetic 
storm,  which  would  otherwise  be  unperceived,  or  from  the 
passage  of  electric  currents  through  the  atmosphere. 

447.  From  the  preceding  articles  it  will  be  apparent  that 


330  VARIATION  OF  THE  COMPASS.  [CHAP.X. 

the  needle,  though  an  invaluable  instrument  for  many  pur- 
poses, is  little  to  be  depended  on  where  precision  is  re- 
quired. It  would  be  very  desirable  that  prominent  marks, 
the  bearings  of  which  were  fully  known,  were  established 
over  the  country,  and  that  all  important  lines  should  be 
determined,  by  triangulation,  from  these.  The  true  bear- 
ings should  always  be  recorded.  There  would  then  be  no 
difficulty  in  retracing  old  lines.  In  the  State  of  Pennsyl- 
vania, and  perhaps  in  some  others,  this  is  now  required  by 
law,  though  it  is  very  doubtful  whether  the  law  is  yet  car- 
ried out  in  a  way  to  be  of  much  practical  benefit,  owing  to 
the  want  of  scientific  knowledge  on  the  part  of  much 
the  larger  number  of  those  who  undertake  the  business  of 
surveying. 

Until  there  is  a  more  general  diffusion  of  theoretical  as 
well  as  practical  science  among  those  whose  business  it  is 
to  settle  the  boundaries  of  estates,  cases  will  continually 
occur  in  which  confusing  lines  will  be  found  to  exist.  This 
could  never  occur  if  all  the  bearings  were  made  to  the  true 
meridian,  the  surveyor  being  careful  to  determine  the  local 
attraction  and  to  allow  for  it  in  making  his  record.  In  no 
instance  should  a  station  be  left  before  the  back-sight  has 
been  taken,  since,  even  in  those  regions  where  but  little 
such  influence  exists,  it  will  sometimes  be  found  at  par- 
ticular points.  It  sometimes  likewise  extends,  without  any 
change,  over  a  considerable  space,  and  thus  may  deflect 
the  needle  similarly  at  a  number  of  stations.  An  instance 
of  this  kind  was  related  to  the  author,  a  short  time  since, 
by  a  surveyor  of  great  practical  experience. 

A  line  was  in  dispute.  One  of  the  parties  called  in  a 
surveyor,  whom  we  shall  call  A.,  who  ran  the  line,  coming 
out  at  a  stone.  The  other  party,  not  being  satisfied,  called 
upon  B.,  who  traced  a  line  agreeing  exactly  with  the  one  run 
by  A.  until  he  came  to  a  certain  point:  he  then  deviated 
from  the  former  line  some  4°  to  the  west.  He  likewise  ar- 
rived at  a  stone.  Both  parties  were  now  dissatisfied.  The  first 
called  on  A.  again,  who  retraced  his  line,  following  exactly 
his  former  course.  B.  was  again  employed.  His  course  de- 
viated at  the  same  point  as  before  from  A.'s.  It  was  then 


VARIATION  OF  THE  COMPASS.  331 

concluded  to  have  them  together.  B.,  being  the  older 
hand,  went  ahead.  When  they  arrived  at  the  point  at 
which  their  lines  separated,  B.  called  on  A.  to  look  through 
the  sights,  saying,  "Is  not  this  right,  Mr.  A.  ?"  "  It  looks 
very  well,"  he  replied:  "but  look  back,  Mr.  B."  On 
doing  so,  he  found  he  was  really  running  4°  to  the  west  of 
his  former  course.  The  attraction  was  first  manifest  at 
that  point,  and  continued,  without  change,  at  all  the  sub- 
sequent stations  along  the  line  he  had  traversed. 


APPENDIX. 


THE  following  demonstration  of  the  rule  for  finding  the  area  of  a  triangle 
when  three  sides  are  given  is  more  concise  than  that  given  in  Art.  251.  As 
the  former,  however,  develops  some  important  properties  respecting  the  centre 
of  the  inscribed  circle,  it  was  thought  best  to  retain  it  :  — 


Fig.  201. 


Let  ABC  (Fig.  201)  be  the  triangle, 
the  construction  being  the  same  as  in 
Fig.  50,  p.  75. 

Then,  as  was  proved  in  the  demon- 
stration of  the  Rule  in  Art.  143, 

AK  =  £  (AB  -f  BC  +  AC)  =  £  s. 
AI  =     «  —  BC. 


We  have  also 

KD  =  Bl  =  £  s  —  AC,  and  KB  =  J  s  —  AB. 
Now,  from  similar  triangles,  ADE  and  AFB,  we  have 
AE  :  ED  :  :  AF  :  FB. 


But 

whence  (23.6) 

But 

and 

and 


AF  :  ED  :  :  AF  :  ED  ; 
AE  .  AF  :  ED2  :  :  AF2  :  ED  .  FB. 
AE  .  AF  =  AK  .  AI  (Cor.  36.3)," 
ED  .  FB  =  HB  .  FB  ==  IB  .  BK  (35.3)  ; 
AI  .  AK  :  EDa  :  :  AF*  :  IB  .  BK, 


.  AK.IB.BK=ED.  AF  =  ED.  (AE-fEF) 
=  ADC  +  BDC  =  ABC. 


332 


MATHEMATICAL  TABLES. 


MATHEMATICAL   TABLES. 


PAGE 

I.  TABLE  OF  LATITUDES  AND  DEPARTURES 3 

II.  TABLE  OF  LOGARITHMS  OF  NUMBERS ,,..,...  17 

III.  TABLE  OF  LOGARITHMIC  SINES  AND  TANGENTS 35 

IV.  TABLE  OF  NATURAL  SINES  AND  COSINES 87 

V.  TABLE  OF  CHORDS 97 


TRAVERSE  TABLE; 


OB, 


DIFFERENCE   OF   LATITUDE 


DEPARTURE. 

/Z/c<J~t-i*'c<sQjX..: 


f*i..'l  1L&> 


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6.9850 
7.9829 
8.9807 
9.9786 

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1.9951 
2.9927 
3.9903 
4.9878 

6.9829 
7.9805 
8.9781 
9.9756 

.0698 

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1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

Dep. 

Lat. 

Dep. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

86f  Deg. 

86*  Deg. 

861  Deg. 

86  Deg. 

22 


LATITUDES  AND  DEPARTURES. 

D. 

41  Deg. 

4i  Deg. 

4|  Deg. 

5  Deg. 

D. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
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1.9945 
2.9918 
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5.9835 
6.9808 
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1.9938 
2.9908 

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6.9784 

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8.9723 
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2.9897 
3.9867 
4.9828 

5-9794 
6.9700 
7.9725 
8.9691 
9.9657 

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1.9924 
2.9886 
3.9848 
4.9810 

5.9772 
6.9734 
7.9696 
8.9658 
9.9619 

.0872 

•1743 
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•5229 
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.6972 

.7844 
.8716 

1 

2  \ 
3 
4 
5 

6 

7 
8 
9 
10 

85|  Deg. 

85£  Deg. 
5£  Deg. 

85i  Deg. 
5f  Deg. 

85  Deg. 

5J  Deg. 

6  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9958 
1.9916 
2.9874 
3.9832 
4.9790 

5.9748 
6.9706 
7.9664 
8.9622 
9.9580 

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1.9908 
2.9862 
3.9816 

4.9770 

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6.9678 
7.9632 
8.9586 
9.9540 

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.7668 
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1.9899 
2.9849 

3-9799 
4.9748 

5.9698 
6.9648 
7-9597 
8-9547 
9-9497 

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.3006 
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.7013 
.8015 
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1.0019 

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1.9890 
2.9836 
3-978i 
4.9726 

5-9671 
6.9617 
7.9562 
8.9507 
9-9452 

.1045 
.2091 
.3136 
.4181 
.5226 

.6272 

:l%l 

.9408 
1-0453 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

841  Deg. 

84J  Deg. 

841  Deg. 
6f  Deg. 

84  Deg. 

61  Deg. 

6J  Deg. 

7  Deg. 

1 
2 
3 
4 

5 

6 

7 
8 
9 
10 

-9941 
1.9881 
2.9822 
3.9762 
4-9703 

5-9643 
6.9584 
7.9524 
8.9465 
9.9406 

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1.0887 

.9936 
1.9871 
2.9807 

3-9743 
4.9679 

5.9614 
6.9550 
7.9486 
8.9421 
9-9357 

.1132 

.2264 
•3396 
•4528 
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.6792 

.7924 
.9056 
1.0188 
1.1320 

•9931 
1.9861 
2.9792 
3-9723 
4-9653 

5-9584 
6.9515 

7-9445 
8.9376 
9.9307 

•"75 
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1.0578 

I-I754 

.9925 
1.9851 

2.9776 
3.9702 
4.9627 

S*55i 

6.9478 

7.9404 
8.9329 
9.9255 

.1219 

•2!3Z 
•3656 

•4875 
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.7312 

•8531 
.9750 
1.0968 

1.2187 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

I 

2 
3 
4 
5 

6 

7 
8 
9 
10 

83f  Deg. 

83  J  Deg. 

83  1  Deg. 

83  Deg. 

71  Deg. 

7J  Deg. 

71  Deg. 

8  Deg. 

.9920 
1.9840 
2.9760 
3.9680 
4.9600 

5.9520 
6.9440 
7.9360 
8.9280 
9.9200 

.1262 
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1.0096 
1.1358 
1.2620 

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1.9829 
2.9743 
3.9658 
4.9572 

5-9487 
6.9401 
7.9316 
8.9230 
9.9144 

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1.0442 
1.1747 
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Lat. 

.9909 
1.9817 
2.9726 
3-9635 
4-9543 

5-9452 
6.9361 
7.9269 
8.9178 
9.9087 

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1.0788 
1.2137 
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1.9805 
2.9708 
3.9611 
4-95I3 
5.9416 
6.9319 
7.9221 
8.9124 
9.9027 

.1392 
.2783 

•4r75 
•5567 
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•8350 
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1.1134 
1.2526 
i-39J7 

Lat. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

D. 

Dep. 

Lat. 

Dep. 

Dep. 

Lat. 

Dep. 

82|  Deg. 

82£  Deg. 

821  Deg. 

82  Deg. 

LATITUDES  AND  DEPARTURES. 

D. 

8i  Deg. 

8i  Deg. 

8f  Deg. 

9  Deg. 

D. 

Lat. 

Dep. 

Lat.     Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

.9897 

1-9793 
2.9690 
3.9586 
4.9483 

5-9379 
6.9276 
7.9172 
8.9069 
9.8965 

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1.0044 
1.1479 
1.2914 
1.4349 

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1.9780 
2.9670 
3-9561 
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5-9341 
6.9231 
7.9121 
8.9011 
9.8902 

.1478 
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.8869 

1.0347 
1.1825 

i-33°3 
1.4781 

.9884 
1.9767 
2.9651 

3-9534 
4.9418 

5-9302 
6.9185 
7.9069 
8.8953 
9.8836 

.1521 

.3042 
.4564 
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1.0649 
1.2170 
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1.5212 

.9877 

1-9754 
2.9631 
3.9508 
4-9384 
5.9261 
6.9138 
7.9015 
8.8892 
9.8769 

.1564 
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.9386 
1.0950 
1.2515 

1.4079 
I-5643 

1 
2  ' 
3  I 
4 
5 

6 

7 
8 
9 
10 

81f  Deg. 

81  J  Deg. 

8H  Deg. 

81  Deg. 

9i  Deg. 

9i  Deg. 

9f  Deg. 

10  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9870 
1.9740 
2.9610 
3.9480 
4-935° 
5.9220 
6.9090 
7.8960 
8.8830 
9.8700 

.1607 

•3215 

.4822 
.6430 
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.9645 
1.1252 
1.2859 
1.4467 
1.6074 

.9863 
1.9726 
2.9589 
3-9451 
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5-9*77 
6.9040 
7.8903 
8.8766 
9.8629 

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.6602 
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I-I553 

1.3204 
1.4854 
1.6505 

.9856 
1.9711 
2.9567 
3.9422 
4.9278 

5-9I33 
6.8989 
7.8844 
8.8700 
9.8556 

.1693 

•3387 
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1.0161 

1.1854 
1-3548 
1.5241 
1-6935 

.9848 
1.9696 
2.9544 
3-9392 
4.9240 

5.9088 

6.8937 
7.8785 
8-8633 
9.8481 

.1736 

•3473 
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.6946 
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1.0419 
1.2155 
1.3892 
1.5628 
1.7365 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

80f  Deg. 

80*  Deg. 

80i  Deg. 

80  Deg. 

10i  Deg. 

10*  Deg. 

10|  Deg. 

11  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9840 
1.9681 
2.9521 
3.9362 
4.9202 

5.9042 
6.8883 
7.8723 
8.8564 
9.8404 

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1.0677 
1.2456 

1-4^35 
1.6015 

1.7794 

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1.9665 
2.9498 
3.9330 
4.9163 

5.8995 
6.8828 
7.8660 
8.8493 
9.8325 

.1822 

•3645 
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1.0934 

1.2756 

1-4579 
1.6401 
1.8224 

.9825 
1.9649 

2-9474 
3.9298 
4.9123 

5-8947 
6.8772 
7.8596 
8.8421 
9-8245 

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1.1191 

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1.4922 
1.6787 
1.8652 

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i-9633 
2-9449 
3.9265 
4.9081 

5.8898 
6.8714 

7-8530 
8.8346 
9.8163 

.1908 
.3816 
.5724 
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1.1449 

1-3357 
1.5265 
1.7173 
1.9081 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

79f  Deg. 

79  *  Deg. 

79i  Deg. 

79  Deg. 

lit  Deg. 

Hi  Deg. 

Hi  Deg. 

12  Deg. 

.9808 
1.9616 
2.9424 

3-923i 
4.9039 

5.8847 
6.8655 
7.8463 
8.8271 
9.8079 

.1951 
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1.1705 
1.3656 
1.5607 
1.7558 
1.9509 

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1.9598 
2.9398 

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4.8996 

5-8795 
6.8595 
7.8394 
8.8193 
9.7992 

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•7975 
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1.1962 
1.3956 
1-5949 
1-7943 
1.9937 

.9790 
1.9581 
2.9371 
3.9162 
4.8952 

5-8743 
6-8533 
7-8324 
8.8114 
9.7905 

.2036 
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1.0182 

1.2219 
1.4255 
1.6291 

1.8328 
2.0364 

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1-95*3 

2-9344 
3.9126 
4.8907 

5.8689 

6.8470 
7.8252 
8.8033 
9-78i5 

.2079 
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1.0396 

1.2475 

1-4554 
1.6633 
1.8712 
2.0791 

Lat. 

1  i 
2 
3 
4 
5 

6 

7  j 
8 
9  ; 
10 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

D. 

78f  Deg. 

78i  Deg. 

78i  Deg. 

78  Deg. 

LATITUDES  AND  DEPARTURES. 

D. 

12i  Deg. 

12*  Deg. 

12|  Deg. 

13  Deg. 

D. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9772 

1-9545 
2.9317 
3.9089 
4.8862 

5.8634 
6.8406 
7.8178 
8.7951 
9.7723 

.2122 
.4244 
.6365 
.8487 
1.0609 

I.273I 
I.4852 
1.6974 
1.9096 
2.I2I8 

.9761 
1.9526 
2.9289 
3.9052 
4.8815 

5.8578 
6.8341 
7.8104 
8.7867 
9.7630 

.2164 
•4329 
•6493 
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1.0822 

1.2986 
1.5151 

i-73J5 
1.9480 
2.1644 

•9753 
1.9507 
2.9260 
3.9014 
4.8767 

5.8521 
6.8274 
7.8027 
8.7781 
9-7534 

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•44  *  4 
.6621 
.8828 
1.1035 

1.3242 
1.5449 
1.7656 
1.9863 

2.2070 

•9744 
1.9487 
2.9231 

3-8975 
4.8719 

5.8462 
6.8206 
7.7950 
8.7693 
9-7437 

.2250 

•4499 
.6749 
.8998 
1.1248 

J-3497 
r-5747 
1.7996 
2.0246 
2.2495 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

2 
3 
4 
5 

6 

7 
8 
9 
-  10 

77f  Deg. 

77*  Deg. 

77i  Deg. 

77  Deg. 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

13i  Deg. 

13*  Deg. 

13  f  Deg. 

14  Deg. 

•9734 
1.9468 
2.9201 

3-8935 
4.8669 

5.8403 
6.8137 
7.7870 
8.7604 
9-7338 

.2292 

•4584 
.6876 
.9168 
1.1460 

1-3752 
1.6044 
1.8336 
2.0628 
2.2920 

•9724 
1.9447 
2.9171 

3-8895 
4.8618 

5-8342 
6.8066 

7.7790 

8-75*3 
9.7237 

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.4669 

.7003 

•9338 
1.1672 

1.4007 
1.6341 
1.8676 

2.IOIO 

2-3345 

•9713 
1.9427 
2.9140 
3-8854 
4-8567 
5.8281 
6.7994 
7.7707 
8.7421 
9-7I34 

.2377 
•4754 
•7I31 
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1.1884 

1.4261 
1.6638 
1.9015 
2.1392 
2.3769 

.9703 
1.9406 
2.9109 
3.8812 
4-8515 
5.8218 
6.7921 
7.7624 
8.7327 
9.7030 

.2419 
.4838 
.7258 
.9677 
1.2096 

1.4515 
I-6935 
1-9354 
2.1773 
2.4192 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

76|  Deg. 

76*  Deg. 

76i  Deg. 

76  Deg. 

14i  Deg. 

14*  Deg. 

14f  Deg. 

15  Deg. 

.9692 
1.9385 

2.9077 
3.8769 
4.8462 

5.8154 
6.7846 
7.7538 
8.7231 
9.6923 

.2462 
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•7385 
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1.2308 

1.4769 
I.723I 
1.9692 
2.2154 
2.4615 

.9681 
1.9363 
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3.8726 
4.8407 

5.8089 
6.7770 
7-7452 
8.7133 
9.6815 

.2504 
.5008 
.7511 
1.0015 
1.2519 

1.5023 

1.7527 
2.0030 
2.2534 
2.5038 

.9670 

1-9341 
2.9011 

3.8682 
4-8352 
5.8023 
6.7693 
7.7364 
8.7034 
9.6705 

.2546 
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1.0184 
1.2730 

1.5276 
1.7822 
2.0368 
2.2914 
2.5460 

•9659 
1.9319 

2.8978 
3.8637 
4.8296 

5-7956 
6.7615 
7.7274 
8.6933 
9-6593 

.2588 
•5*76 
•7765 
1-0353 
1.2941 

1.5529 
1.8117 
2.0706 
2.3294 
2.5882 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

75f  Deg. 

75*  Deg. 

75i  Deg. 

75  Deg. 
16  Deg. 

15i  Deg. 

15*  Deg. 

15f  Deg. 

.9648 
1.9296 
2.8944 

3-8591 
4.8239 

5-7887 
6-7535 
7.7183 
8.6831 
9.6479 

.2630 
.5261 
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I.052I 
I.3I52 

1.5782 
I.84I2 
2.1042 

2.3673 
2.630^ 

.9636 
1.9273 
2.8909 

3-8545 
4.8182 

5.7818 

6.7454 
7.7090 

8-6727 
9-6363 

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•5345 
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1.0690 
1.3362 

1.6034 
1.8707 
2.1379 
2.4051 
2.6724 

.9625 
1.9249 

2.8874 

3-8498 
4.8123 

5-7747 
6.7372 
7.6996 
8.6621 
9.6246 

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.5429 
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1.0858 
1-3572 
1.6286 
1.9001 
2.1715 
2.4430 
2.7144 

.9613 
1.9225 
2.8838 

3-8450 
4.8063 

7.6901 

8-6514 
9.6120 

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.8269 
1.1025 
1.3782 

1-6538 
1.9295 
2.2051 
2.4807 
2.7564 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

74f  Deg. 

74*  Deg. 

74i  Deg. 

74  Deg. 

LATITUDES  AND  DEPARTURES. 

D. 

161  Deg. 

16*  Deg. 

16f  Deg. 

17  Deg. 

D. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

.9600 
1.9201 

2.8801 
3.8402 
4.8002 

5.7603 
6.7203 
7.6804 
8.6404 
9.6005 

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1.1193 

I-3991 
1.6790 
1.9588 
2.2386 
2.5185 
2.7983 

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1.9176 
2.8765 

3-8353 
4.7941 

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6.7117 
7.6706 
8.6294 
9.5882 

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1.1361 
1.4201 

1.7041 
1.9881 
2.2721 
2.5561 
2.8402 

.9576 
1.9151 

2.8727 

3-8303 
4.7879 

5-7454 
6.7030 
7.6606 
8.6181 
9-5757 

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1.1528 
1.4410 

1.7292 
2.0174 
2.3056 
2.5938 
2.8820 

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1.9126 
2.8689 

3.8252 
4.7815 

5.7378 

6.6941 

7.6504 

8.6067 
9.5630 

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1.1695 
1.4619 

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2.0466 
2.3390 
2.6313 
2.9237 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

73f  Deg. 

73  £  Deg. 

73*  Deg. 

73  Deg. 

17*  Deg. 

17£  Deg. 

17f  Deg. 

18  Deg. 

.9550 
1.9100 
2.8651 
3.8201 
4-7751 

5-7301 
6.6851 
7.6402 
8.5952 
9.5502 

.2965 

^8896 
1.1862 
1.4827 

1.7792 
2.0758 
2.3723 
2.6689 
2.9654 

•9537 
1.9074 
2.8612 
3.8149 
4.7686 

H**3 
6.6700 
7.629} 

8-5835 
9.5372 

.3007 
.6014 
.9021 
1.2028 
I-5035 
1.8042 
2.1049 
2.4056 
2.7064 
3.0071 

•95*4 
1.9048 
2.8572 
3.8096 
4.7620 

5-7I44 
6.6668 
7.6192 
8.5716 
9.5240 

•3°49 
.6097 
.9146 
1.2195 
i-5*43 
1.8292 
2.134! 
2.4389 
2.7438 
3.0486 

•95" 
1.9021 
2.8532 
3.8042 
4-7553 
5.7063 
6.6574 
7.6085 

8-5595 
9.5106 

.3090 
.6180 
.9271 
1.2361 
1.5451 

1.8541 
2.1631 

2.4721 
2.7812 
3.0902 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

72f  Deg. 

72*  Deg. 

72*  Deg. 

72  Deg. 

18*  Deg. 

18J  Deg. 

18f  Deg. 

19  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

•9497 
1.8994 
2.8491 
3.7988 
4.7485 

5.6982 
6.6479 
7-5976 

8-5473 
9.4970 

.6263 

•9395 

1.2527 
1.5658 

1.8790 
2.1921 

*-5°53 
2.8185 
3.1316 

-9483 
1.8966 
2.8450 

3-7933 
4.7416 

5.6899 
6.6383 
7.5866 
8.5349 
9.4832 

•3173 
.6346 
.9519 
1.2692 
1.5865 

1.9038 
2.221  1 

2.5384 
2.8557 
3.1730 

.9469 

1-8939 

2.8408 

3-7877 
4-7347 
5.6816 
6.6285 

7-5754 
8.5224 

9-4693 

.3214 
.6429 
.9643 
1.2858 
1.6072 

1.9286 
2.2501 

2.5715 
2.8930 
3.2144 

•9455 
1.8910 
2.8366 
3.7821 
4.7276 

5-673T 
6.6186 
7.5641 
8.5097 
9-455* 

.3256 
.6511 
.9767 
1.3023 
1.6278 

1-9534 
2.2790 
2.6045 
2.9301 

3-*557 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

71|  Deg. 

71i  Deg. 

71*  Deg. 

71  Deg. 

191  Deg. 

19  j-  Deg. 

19|  Deg. 

20  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

.9441 
1.8882 
2.8323 

3-7764 
4.7204 

5.6645 
6.6086 

7-55*7 
8.4968 
9.4409 

•3*97 
.6594 
.9891 
1.3188 
1.6485 

1.9781 

2.3078 
2.6375 
2.9672 
3.2969 

.9426 
1.8853 
2.8279 
3.7706 
4-713* 

5.6558 
6.5985 
7.5411 
8.4838 
9.4264 

•3338 
.6676 
I.OOI4 

'IP 
1.6690 

2.0028 
2.3366 
2.6705 
3.0043 
3-338I 

.9412 
1.8824 
2.8235 
3.7647 
4.7059 

5.6471 

7.5294 
8.4706 
9.4118 

•3379 
.6758 
1.0138 

'•SS1? 
1.6896 

2.0275 
2.3654 
2.7033 
3.0413 
3-379* 

•9397 
1.8794 
2.8191 

3-7588 
4.6985 

5.6382 
6.5778 
7-5J75 
8-457* 
9-3969 

•%& 

1.0261 
1.3681 
1.7101 

2.0521 
2.3941 
2.7362 
3.0782 
3.4202 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

70f  Deg. 

70*  Deg. 

70*  Deg. 

70  Deg. 

LATITUDES  AITO  DEPARTURES. 

D. 

20*  Deg. 

20*  Deg. 

20f  Deg. 

21  Deg. 

D. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

.9382 
1.8764 
2.8146 
3-75^8 
4.6910 

5.6291 
6.5673 
7-5°55 
8-4437 
9.3819 

.3461 
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1.0384] 
1.3845 
1.7306 

2.0767 
2.4228 
2.7689 

3-"5i 
3.4612 

•9367 

1-8733 

2.8100 
3.7467 
4.6834 

5.6200 
6.5567 

7-4934 
8.4300 
9.3667 

.3502 

.7004  ! 
1.0506 
1.4008 
1.7510 

2.IOI2 

2.4515 
2.8017 

1-I5*9 
3.5021 

•9351 

1.8703 
2.8054 
3-74°5 
4-6757 
5.6108 

6-5459 
7.4811 
8.4162 
9-35H 

•3543 
.7086 
1.0629 
1.4172 
1-77*5 
2.1257 
2.4800 
2.8343 
3.1886 
3-5429 

•9336 

1.8672 
2.8007 

3-7343 
4.6679 

5.6015 

6-5351 
7.4686 
8.4022 
9-3358 

•3584 
.7167 
1.0751 

J-4335 
1.7918 

2.1502 
2.5086 
2.8669 
3.2253 
3-5837 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

1 
2 
3 
4 
5 

6 

7 
8  i 
9 
10 

69|  Deg. 

69  J  Deg. 
21  J  Deg. 

69i  Deg. 

69  Deg. 

21  1  Deg. 

21f  Deg. 

22  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9320 
1.8640 
2.7960 
3.7280 
4.6600 

5-592° 
6.5241 
7-456i 
8.3881 
9.3201 

.3624 

.7249  1 
1.0873 
1.4498 
1.8122 

2.1746 

2-5371 
2.8995 
3.2619] 

3.6244 

•93°4 
i.  8608 
2.7913 
3.7217 
4.6521 

5.5825 
6.5129 

7-4433 
8.3738 
9.3042 

.3665 

•733° 
1.0995 
1.4660 
1.8325 

2.1990 

2-5655 
2.9320; 
3.2985 
3.6650 

.9288 
1.8576 
2.7864 

3-7I52 
4.6440 

5-5729 

6.5017 
7.4305 

9.2881 

.3706 
.7411 
1.1117 

1.4822 
1.8528 

2.2233 

2-5939 
1.9645 

3-335° 
3.7056 

.9272 

1.8544 
2.7816 
3.7087 
4-6359 

5-5631 
6.4903 

7-4*75 
8-3447 
9.2718 

•3746 
.7492 
1.1238 
1.4984 
1.8730 

2.2476 
2.6222 
2.9969 

3-37I5 
3.7461 

68f  Deg. 

68  J  Deg. 

68i  Deg. 

68  Deg.  . 

~T| 

2 
3 
4 
5 

6 

7 
8 
9 
1O 

22i  Deg. 

22  £  Deg. 

22f  Deg. 

23  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

.9255 
1.8511 

2.7766 
3.7022 
4.6277 

8£i 

7.4043 
8.3299 

9-2554 

.3786 

•7573 
1.1359 
1.5146 
1.8932 

2.2719 
2.6505 
3.0292 

3.4078 
3-7865 

.9239 

1.8478 
2.7716 

3-6955 
4.6194 

5-5433 
6.4672 
7.3910 
8.3149 
9.2388 

.3827 

•7654 
1.1481 

i-53°7 
I-9J34 
2.2961 
2.6788 
3.0615 
3.4442 
3.8268 

.9222 
1.8444 
2.7666 
3.6888 
4.6110 

5-5332 
6-4554 
7.3776 
8.2998 
9.2220 

.3867 
•7734 

I.IOOI 

1.5468 
I-9336 

2.3203 
2.7070 
3.0937 
3.4804 
3.8671 

.9205 
1.8410 
2.7615 
3.6820 
4.6025 

5-523° 
6-4435 
7.3640 
8.2845 
9.2050 

•39°7 
.7815 
1.1722 
1.5629 
J-9537 
2.3444 
2-7351 

3'12^ 
3.5166 

3.9073 

671  Deg. 
23i  Deg. 

67  i  Deg. 

67i  Deg. 

67  Deg. 

23  £  Deg. 

23|  Deg. 

24  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9188 
1.8376 
2.7564 
3.6752 
4-594° 

5-5I27 
6-43  T  5 
7-3503 
8.2691 
9.1879 

•3947 
.7895 
1.1842 
1.5790 
1-9737 
2.3685 
2.7632 
3-158° 
3-55^7 
3-9474 

.9171 
1.8341 

2.7512 
3.6682 
4-5853 
5.5024 
6.4194 

7-3365 
8.2535 
9.1706 

•3987 
•7975 
1.1962 
1.5950 
1.9937 

2.3925 
2.7912 
3.1900 
3-5887 
3-9875 

'%I5l 
1.8306 

2.7459 
3.6612 
4-5766 

5-49*9 

6.4072 

7-3225 
8.2378 

1  9-J531 

.4027 
.8055 
1.2082 
1.6110 

2.0137 

2.4165 
2.8192 
3.2220 
3.6247 
4.0275 

•9*35 

1.8271 
2.7406 
3-6542 
4-5677 

5'48i3 
6.3948 
7.3084 
8.2219 
9-'355 

.4067 
•8i35 

I.22O2 
1.6269 

2.0337 

2.4404 
2.8472 

3-2539 
3.6606 
4.0674 

1 
2 
3 
4 
5 

6 

7 
8  i 
9 
10 

D. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

66f  Deg. 

66J  Deg. 

66i  Deg. 

66  Deg. 

10 


_  —  .  -               -. 

LATITUDES  AND  DEPARTURES. 

i  D- 

24i  Deg. 

24£  Deg. 

24f  Deg. 

25  Deg. 

D. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

.9063 
1.8126 
2.7189 
3.6252 
4'53!5 

5-4378 
6.3442 

7-2505 
8.1568 
9.0631 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9118 
1.8235 

2-7353 
3.6470 
4.5588 

5.4706 
6.3823 
7.2941 
8.2059 
9.1176 

.4107 
.8214 
1.2322 
1.6429 
2.0536 

2.4643 
2.8750 
3.2858 
3.6965 

4.1072 

.9100 
1.8199 
2.7299 
3.6398 
4.5498 

5-4598 
6.3697 
7.2797 
8.1897 
9.0996 

•4I47| 
.82945 
1.2441 
1.6588 
2.0735 

2.4882 
2.9029 

3-3'75 
3-7322 
4.1469 

.9081 
1.8163 
2.7244 
3.6326 
4.5407 

5.4489 
6.3570 
7.2651 

8-1733 
9.0814 

.4187 

•8373 
1.2560 
1.6746 
2.0933 

2.5120 
2.9306 

3-3493 
3.7679 
4.1866 

.4226 
.8452 
1.2679 
1.6905 
2.1131 

2-5357 
2.9583 
3.3809 
3.8036 
4.2262 

65f  Deg. 

65  \  Deg. 
25  £  Deg. 

65i  Deg. 
25f  Deg. 

65  Deg. 

25i  Deg. 

26  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.9045 
1.8089 
2.7134 
3.61/8 
4.5223 

5.4267 
6.3312 
7.2356 
8.1401 
9.0446 

.4266 
.8531 
1.2797 
1.7063 
2.1328 

2-5594 
2.9860 

3-4^5 

3-839I 
4.2657 

.9026 

1.8052 
2.7078 
3.6103 
4.5129 

5-4I55 
6.3181 
7.2207 
8.1233 
9.0259 

•43°5 
.8610 
1.2915 
1.7220 
2.1526 

2.5831 
3.0136 

3-4441 
3.8746 
4.3051 

.9007 
1.8014 
2.7021 
3.6028 
4-5°35 
5.4042 
6.3049 
7.2056 
8.1063 
9.0070 

•4344 
.8689 

I-3033 

1.7378 
2.1722 

2.6067 
3.0411 

3-4756 
3.9100 

4-3445 

.8988 
1.7976 
2.6964 

3-5952 
4.4940 

5.3928 
6.2916 
7.1904 
8.0891 
8.9879 

.4384 
.8767 

I-3J51 
1-7535 
2.1919 

2.6302 
3.0686 
3.5070 
3-9453 
4-3837 

1 
2 
3 
4 
5 

6 

7. 
8 
9 
10 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
1O 

64f  Deg. 

64£  Deg. 

64i  Deg. 

64  Deg. 

26i  Deg. 

26£  Deg. 

26f  Deg. 

27  Deg. 

.8969 

1.7937 
2.6906 

3-5^75 
4.4844 

5.3812 
6.2781 
7.1750 
8.0719 
8.9687 

•4423 
.8846 
1.3269 
1.7692 
2.2114 

2.6537 
3.0960 
3-5383 
3.9806 
4.4229 

.8949 

1.7899 
2.6848 
3-5797 
4-4747 
5.3696 
6.2645 

7-1595 
8.0544 

8-9493 

.4462 
.8924 
1.3386 
1.7848 
2.2310 

2.6772 

3>If3t 
3.5696 

4.0158 
4.4620 

.8930 
1.7860 
2.6789 

3-5719 
4.4649 

5-3579 
6.2509 
7.1438 
8.0368 
8.9298 

.4501 
.9002 

I-3503 
1.8004 
2.2505 

2.7006 

3-I5°7 
3.6008 
4.0509 
4.5010 

.8910 

1.7820 
2.6730 
3.5640 
4.4550 

5.3460 
6.2370 
7.1281 
8.0191 
8.9101 

•4540 
.9080 
1.3620 
i.  8160 
2.2700 

2.7239 

3-1779 
3.6319 
4.0859 
4-5399 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

63|  Deg. 

63  i  Deg. 

63i  Deg. 

63  Deg. 

27i  Deg. 

27£  Deg. 

271  Deg. 

28  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.8890 
1.7780 
2.6671 
3-556i 
4.4451 

5-3341 
6.2231 
7.1121 

8.0012 

8.8902 

•4579 
•9J57 
1.3736 
1.8315 
2.2894 

2.7472 

3-2°5' 
3.6630 
4.1209 

4-5787 

.8870 

'i!40 

2.6610 

3.5480 

4-43  5  1 
5.3221 
6.2091 
7.0961 

8.8701 

.4617 

•9235 
1.3852 
1.8470 
2.3087 

2.7705 
3.2322 
3.6940 

4-1557 
4.6175 

.8850 
1.7700 
2.6550 
3.5400 
4.4249 

5-3°99 
6.1949 
7.0799 
7.9649 
8.8499 

.4656 
.9312 
1.3968 
1.8625 
2.3281 

2.7937 

3-2593 
3.7249 
4.1905 
4.6561 

.8829 
1.7659 
2.6488 
3-53i8 
4.4147 

5.2977 
6.1806 
7.0636 
7.9465 
8.8295 

.4695 
.9389 

1.4084 
1.8779 
2-3474 
2.8168 
3.2863 
3-7558 
4.2252 
4.6947 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

D. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

62f  Deg. 

62  }  Deg. 

62i  Deg. 

62  Deg. 

11 


LATITUDES  AIT3 

D  DEPART 

CJKZSS. 

D. 

28i  Peg. 

28J  Deg. 

28f  Deg. 

29  Deg. 

D. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

.8809 
1.7618 
2.6427 
3-5236 

4.4045 

l^l3 
6.1662 

7.0471 
7.9280 
8.8089 

•4733 
.9466 
1.4200 

'•8933 
2.3666 

2.8399 
3-3132 
3.7866 
4.2599 
4.7332 

.8788 

1'1S16 
2.6365 

3-5I53 
4.3941 

5-2729 
6.1517 
7.0305 
7.9094 
8.7882 

.4772 
•9543 
I-43I5 
1.9086 
2.3858 

2.8630 
3.3401 

3-8i73 
4.2944 
4.7716 

.8767 

1-7535 
2.6302 
3.5069 
4-3836 
5.2604 
6.1371 
7.0138 
7.8905 
8.7673 

.4810 
.9620 
1.4430 
1.9240 
2.4049 

2.8859 
3.3669 
3.8479 
4-3289 
4.8099 

.8746 
1.7492 
2.6239 
3-4985 
4-3731 

5-2477 
6.1223 
6.9970 
7.8716 
8.7462 

.4848 
.9696 
1.4544 
1.9392 
2.4240 

2.9089 
3-3937 
3-8785 
4-3633 
4.8481 

~~T 

2 
3 

i  4 
5 

6 

7 
8 
9 
10 

61f  Deg. 

61  £  Deg. 

61  i  Deg. 

61  Deg. 
30  Deg. 

29J  Deg. 

29*  Deg. 

29|  Deg. 

.8725 
1.7450 
2.6175 
3.4900 
4.3625 

5-235o 
6.1075 
6.9800 
7.8525 
8.7250 

.4886 
.9772 
1.4659 

1-9545 

2.4431 

2.9317 
3.4203 
3.9090 
4.3976 
4.8862 

.8704 
1.7407 

2.6lH 

3.4814 
4.3518 

5.2221 
6.0925 

6.9628 
7-8332 
8.7036 

.4924 
.9848 

J-4773 
1.9697 
2.4621 

2-9545 
3-4470 
3-9394 
4.4318 
4.9242 

.8682 
1.7364 
2.6046 
3-4728 
4.3410 

5.2092 
6.0774 
6.9456 
7.8138 
8.6820 

.4962 
.9924 
1.4886 
1.9849 
2.4811 

2.9773 
3-4735 
3-9697 
4.4659 
4.9622 

.8660 
1.7321 
2.5981 
3.4641 
4-3301 
5.1962 
6.0622 
6.9282 

EBS 

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I.OOOO 

1.5000 

2.OOOO 

2.5000 
3.0000 

3.5000 

4.0000 

4.5000 

5.0000 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

60f  Deg. 

60  £  Deg. 

60i  Deg. 

60  Deg. 

30  i  Deg. 

30  £  Deg. 

30|  Deg. 

31  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.8638 
1.7277 
2.5915 

3-4553 
4.3192 

5.1830 
6.0468 
6.9107 

7-7745 
8.6384 

.5038 
1.0075 

*-5"3 
2.0151 
2.5189 

3.0226 
3.5264 
4.0302 
4.5340 
5.0377 

.8616 

1.7233 
2.5849 

3-4465 
4.3081 

5.1698 
6.0314 
6.8930 

7-7547 
8.6163 

•5°75 
1.0151 

1.5226 
2.0302 
2-5377 
3.0452 
3-5528 
4.0603 
4-5678 
5-0754 

.8594 

1.7188 
2.5782 
3-4376 
4.2970 

5.1564 
6.0158 
6.8753 
7-7347 
8.5941 

•5"3 
1.0226 

1-5339 
2.0452 

2-5565 
3.0678 

3-5791 
4.0903 
4.6016 
5.1129 

.8572 
i»7*43 

2.5715 
3.4287 
4.2858 

5-143° 

6.OOO2 

6.8573 
7-7145 
8-5717 

.5150 

1.0301 

1-5451 

2.0602 

2.5752 

3.0902 
3.6053 
4.1203 
4-6353 
5^504 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

59|  Deg. 

59*  Deg. 
3H  Deg. 

59i  Deg. 

59  Deg. 

3H  Deg. 

31f  Deg. 

32  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.8549 
1.7098 
2.5647 
3.4196 
4.2746 

5-i295 
5.9844 
6.8393 
7.6942 
8.5491 

.5188 
1-0375 
i-5563 
2.0751 

2-5939 
3.1126 

3-63i4 
4.1502 
4.6690 
5-1877 

.8526 

1-7053 
2-5579 
3.4106 
4.2632 

5.1158 
5.9685 
6.8211 
7.6738 
8.5264 

.5225 

1.0450 

1-5675 
2.0900 
2.6125 

3-I350 

3-6575 
4.1800 
4.7025 
5.2250 

.8504 

1.7007 
2.5511 
3.4014 
4.2518 

5.1021 

5-9525 
6.8028 
7.6532 
8.5035 

.5262 
1.0524 
1.5786 
2.1049 
2.6311 

3-'573 
3-6835 
4.2097 

4-7359 
5.2621 

I  .8480 
1.6961 
2.5441 
3.3922 
4.2402 

5-0883 

5-9363 
6.7844 

7-6324 
8.4805 

.5299 

1.0598 

1.5898 
*"1191 

2.6496 

3-J795 
3.7094 
4.2394 

4-7693 
5.2992 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

D. 

Dep. 

Laf. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

58f  Deg. 

58*  Deg. 

58  1  Deg. 

58  Deg. 

12 


LATITUDES  AND  DEPARTURES. 

D. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

321  Deg. 

32*  Deg. 

32f  Deg. 

33  Deg. 

D. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

.8457 
1.6915 
2.5372 
3.3829 
4.2286 

5.0744 
5.9201 
6.7658 
7.6116 
8-4573 

.5336 
1.0672 
1.6008 

2.1345 
2.6681 

3.2017 

3-7353 
4.2689 
4.8025 
5.336i 

.8434 
1.6868 
2.5302 

3-3736 
4.2170 

5.0603 

5-9°37 
6.7471 

7'59°5 
8.4339 

•5373 
1.0746 
1.6119 
2.1492 
2.6865 

3.2238 
3-76ii 
4.2984 

4-8357 
5.3730 

.8410 
1.6821 
2.5231 
3.3642 
4.2052 

5.0462 
5-8873 
6.7283 
7.5694 
8.4104 

.5410 
1.0819 
1.6229 
2.1639 
2.7049 

3-2458 
3.7868 

4-3278 
4.8688 
5.4097 

-8387 
1.6773 
2.5160 
3-3547 
4-1934 
5.0320 
5.8707 
6.7094 
7.5480 
8.3867 

.5446 
1.0893 
1.6339 
2.1786 
2.7232 

3.2678 
3.8125 

4-3571 
4.9018 

5-4464 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

57i  Deg. 

57*  Deg. 

571  Deg. 

57  Deg. 

331  Deg. 

33*  Deg. 

33f  Deg. 

34  Deg. 

•8363 
1.6726 
2.5089 

3-3451 
4.1814 

5.0177 
5.8540 
6.6903 
7.5266 
8.3629 

•5483 
1.0966 
1.6449 
2.1932 
2.7415 

3.2898 
3.8381 
4.3863 
4.9346 
5.4829 

.8339 

1.6678 
2.5017 

3-3355 
4.1694 

5-°°33 
5-837* 
6.6711 

7-5°5° 
8-3389 

•55i9 
1.1039 
1.6558 
2.2077 
2.7597 

3.3116 
3.8636 
4-4I55 
4-9674 
5-5*94 

•8315 
1.6629 
2.4944 
3-3259 
4-1573 
4.9888 
5.8203 
6.6518 
7.4832 
8.3147 

•5556 
I.IIII 

1.6667 
2.2223 
2.7779 

3-3334 
3.8890 
4.4446 
5.0001 
5-5557 

.8290 
1.6581 
2.4871 
3.3162 
4.1452 

4.9742 

5-8033 
6.6323 
7.4613 
8.2904 

•5592 
1.1184 

1.6776 
2.2368 
2.7960 

3-3552 
3.9144 

4-4735 
5.0327 

5-59I9 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

56|  Deg. 

56*  Deg. 

561  Deg. 

56  Deg. 

341  Deg. 

34*  Deg. 

34|  Deg. 

35  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

.8266 
1.6532 
2.4798 
3.3064 
4.1329 

4-9595 
5.7861 
6.6127 

7-4393 
8.2659 

.5628 
1.1256 
1.6884 
2.2512 
2.8140 

3.3768 
3.9396 

4.5024 
5.0652 
5.6280 

.8241 
1.6483 
2.4724 
3.2965 
4.1206 

4.9448 
5.7689 

6-593° 

7.4171 
8.2413 

.5664 
1.1328 
1.6992 
2.2656 

2.8320 

3-3984 
3.9648 
4-53*2 
5.0977 
5.6641 

.8216 

1-6433 
2.4649 
3.2866 
4.1082 

4.9299 

5-75I5 
6.5732 
7.3948 
8.2165 

.5700 
1.1400 
1.7100 
2.2800 
2.8500 

3.4200 
3.9900 
4.5600 
5.1300 
5.7000 

.8192 
1.6383 

2-4575 
3.2766 
4.0958 

4.9149 
5-7341 
6.5532 
7-3724 
8.1915 

.5736 
1.1472 
1.7207 
2.2943 
2.8679 

3-44I5 
4.0150 
4.5886 
5.1622 
5-7358 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

55|  Deg. 

55*  Deg. 

551  Deg. 
351  Deg. 

55  Deg. 

351  Deg. 

353  Deg. 

36  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

.8166 
1-6333 

2'41^ 
3.2666 

4.0832 

4.8998 
5-7165 

6-5331 
7.3498 
8.1664 

•5771 
i-I543 
i-73J4 
2.3086 
2.8857 

3.4629 

4.0400 
4.6172 
5-'943 

5-77I5 

.8141 

1.6282 
2.4423 

3-^565 
4.0706 

4.8847 
5.6988 
6.5129 
7.3270 
8.1412 

.5807 
1.1614 

1.7421 
2.3228 
2.9035 

3.4842 
4.0649 
4.6456 
5.2263 
5.8070 

.8116 
1.6231 
2-4347 
3-2463 
4.0579 

4.8694 
5.6810 
6.4926 
7.3042 
8.1157 

.5842 
1.1685 
1.7527 
2.3370 
2.9212 

3-5055 
4.0897 
4.6740 
5.2582 
5.8425 

.8090 

1.6180 
2.4271 
3.2361 
4.0451 

4.8541 
5.6631 
6.4721 
7.2812 
8.0902 

•5878 
1.1756 
1.7634 
2.3511 
2.9389 

3.5267 
4.1145 
4.7023 
5.2901 
5-8779 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

1 

Dep. 

Lat. 

D. 

54i  Deg.          54*  Deg. 

i     541  Deg. 

54  Deg. 

LATITUDES  AND  DEPARTURES. 

D. 

1 
2 
3 
4 
5 

6 

7 
8 
i     9 
10 

~~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

361:  Deg. 

361  Deg. 

36f  Deg. 

37  Deg. 

D. 

1 
2 
3 
4 
5 

6 

7 
S 
9 
10 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

.8064 
1.6129 
2.4193 
3.2258 
4.0322 

4-8387 
5-645I 
6.4516 

7.2580 
8.0644 

•59'3 

1.1826 

1-7739 
2.3652 

2-9565 

3-5479 
4.1392 

4-7305 
5.3218 

5-9J31 

.8039 

1.6077 
2.4116 

3-2154 
4.0193 

4.8231 
5.6270 
6.4309 

7-2347 
8.0386 

•5948 
1.1896 
1.7845 
2-3793 
2-9741 

3-5689 
4.1638 
4.7586 

5-3534 
5.9482 

.8013 

1.6025 
2.4038 
3-2050 
4.0063 

4.8075 
5.6088 
6.4100 
7.2113 
8.0125 

•5983 
1.1966- 
1.7950 

2-3933 
2.9916 

3-5899 
4.1883 
4.7866 
5-3849 
5-9832 

.7986 

'•5973 
2.3959 

3-1945 
3.9932 

4.7918 

5-5904 
6.3891 
7.1877 
7.9864 

.6018 
1.2036 
1.8054 

2.4°73 
3.0091 

3.6109 

4.2127 
4.8145 

5-4l63 
6.0181 

53f  Deg. 

53^  Deg. 

53  i  Deg. 

53  Deg. 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

37i  Deg. 

371  Deg. 

371  Deg. 

38  Deg. 

.7960 
1.5920 
2.3880 
3.1840 
3.9800 

4.7760 

5-57*° 
6.3680 
7.1640 
7.9600 

•6053, 
i.  2106 
1.8159 

2.4212 
3.0265 

3-6318 
4.2371 
4.8424 
5-4476 
6.0529 

•7934 
1.5867 
2.3801 

3>IHi 
3.9668 

4.7601 

5-5535 
6.3468 
7.1402 
7-9335 

.6088 
1.2175 
1.8263 
2.4350 
3-0438 
3.6526 
4.2613 
4.8701 

5-4789 
6.0876 

.7907 
1.5814 
2.3721 
3.1628 
3-9534 
4.7441 
5-5348 
6-3255 
7.1162 
7.9069 

.6122 

1.2244 
1.8367 
2.4489 
3.0611 

3-6733 
4.2855 

4-8977 
5.5100 

6.1222 

.7880 
1.5760 
2.3640 
3.1520 
3.9401 

4.7281 
5.5161 
6.3041 
7.0921 

7.8801 

.6157 

1-2313 

1.8470 
2.4626 
3.0783 

3.6940 
4.3096 
4-9253 

I'54^ 
6.1566 

52|  Deg. 

521  Deg. 

52i  Deg. 

52  Deg. 
39  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

38i  Deg. 

381  Deg. 

38f  Deg. 

1 
,  2 
3 
4 
5 

6 

7 
8 
9 
10 

•7853 
1.5706 
2.3560 

3'14il 
3.9266 

4.7119 

5-4972 
6.2825 
7.0679 
7-8532 

.6191 
1.2382 
1.8573 
2.4764 
3-0955 
3-7I46 
4-3337 
4.9528 
5.5718 
6.1909 

.7826 
1.5652 
2.3478 
3.1304 
3.9130 

4.6956 

5-4783 
6.2609 
7.0435 
7.8261 

.6225 

1.2450 
1.8675 
2.4901 
3.1126 

3-7351 
4-3576 
4.9801 
5.6026 
6.2251 

•7799 
1.5598 

2.3397 
3-"95 
3.8994 

4.6793 
5-4592 
6.2391 
7.0190 
7-7988 

.6259 
1.2518 
1.8778 

2.5037 
3.1296 

3-7555 
4-3815 
5-oo74 
5-6333 
6.2592 

.7771 
1-5543 
2-33H 
3.1086 

3-8857 
4.6629 
5.4400 
6.2172 
6-9943 
7-77I5 

.6293 

1.2586 
1.8880 

2-5'73 
3.1466 

3-7759 

4.4052 
5.0346 
5.6639 
6.2932 

51|  Deg. 
39i  Deg. 

511  Deg. 

5H  Deg. 

51  Deg. 

391  Deg. 

39f  Deg. 

40  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

•7744 
1.5488 
2.3232 
3.0976 
3.8720 

4.6464 
5.4207 

*-19:51 
6.9695 

7-7439 

.6327 
1.2654 
1.8981 
2.5308 
3-l635 
3.7962 
4.4289 
5.0616 

5-6943 
6.3271 

.7716 
1-5432 
2.3149 
3.0865 
3.8581 

4.6297 
5.4014 
6.1730 
6.9440 

7.7162 

Dep. 

.6361 

1.2722 
1.9082 

2-5443 
3.1804 

3.8165 

4.4525 
5.0886 

5-7247 
6.3608 

.7688 

i-5"377 
2.3065 

3-0754 
3.8442 

4.6131 

5-38i9 
6.1507 
6.9196 
7.6884 

•6394 
1.2789 
1.9183 
.2-5578 
3.1972 

3.8366 
4.4761 
5-"55 
5-755° 
6-3944 

.7660 
1.5321 
2.2981 
3.0642 
3.8302 

4-5963 
5-3623 
6.1284 
6.8944 
7.6604 

.6428 
1.2856 
1.9284 
2.5712 
3.2139 

3-8567 
4-4995 
5-i423 
5-7851 
6.4279 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

D. 

Dep. 

Lat. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D 

50|  Deg. 

501  Deg. 

50J  Deg. 

50  Deg. 

LATITUDES  AND  DEPARTURES. 

D. 

40i  Deg. 

40£  Deg. 

40f  Deg. 

41  Deg. 

D. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  - 

Dep. 

Lat. 

Dep. 

1 
2 

1    3 
4 
5 

6 

7 
8 
9 
10 

.7632 
1.5265 
2.2897 
3.0529 
3.8162 

4-5794 
5.3426 
6.1059 
6.8691 
7.6323 

.6461 
1.2922 
1.9384 
2.5845 
3.2306 

3.8767 
4.5229 
5.1690 
5.8151 
6.4612 

.7604 
1.5208 
2.2812 
3.0416 
3.8020 

4.5624 
5.3228 
6.0832 
6.8437 
7.6041 

.6494 
1.2989 
1.9483 
2.5978 
3-2472 
3.8967 
4.5461 

5-I956 
5.8450 
6.4945 

.7576 
1.5151 

2.2727 
3.0303 
3-7878 

4-5454 
5-3030 
6.0605 
6.8181 
7-5756 

.6528 
1-3055 
1-9583 

2.6110 

3.2638 

3.9166 

4-5693 

5.2221 

5.8748 

6.5276 

•7547 
1.5094 
2.2641 
3.0188 
3-7735 

4-5283 
5.2830 
6.0377 
6.7924 
7-5471 

.6561 
1.3121 

1.9682 
2.6242 
3-2803 

3-9364 
4.5924 
5.2485 

5-9045 
6.5606 

~T 

!     2 
3 
,     4 
5 

6 

7 
8 
9 
10 

49|  Deg. 

49  J  Deg. 
41  J  Deg.  ~ 

49i  Deg. 

49  Deg. 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

4H  Deg. 

41f  Deg. 

42  Deg. 

.7518 
i-S«>37 

2-^555 
3.0074 

S-7592 
4.5110 
5.2629 
6.0147 
6.7666 
7.5184 

•6593 
1.3187 
1.9780 
2.6374 
3.2967 

3-956I 
4.6154 
5.2748 
5-9341 
6-5935 

.7490 
1.4979 
2.2469 
2.9958 
3-7448 

4-4937 
5.2427 
5.9916 
6.7406 
7.4896 

.6626 
1.3252 
1.9879 
2.6505 
3-3I31 

3-9757 
4.6383 

5-3^ 
5.9636 

6.6262 

.7461 
1.4921 
2.2382 
2.9842 
3-7303 
4.4763 
5.2224 
5.9685 

6'7'45 
7.4606 

.6659 
1.3318 
1.9976 
2.6635 
3-3294 

3-9953 
4.6612 

5-327I 
5-9929 
6.6588 

•7431 
1.4863 
2.2294 
2.9726 
3-7I57 
4.4589 
5.2020 
5-9452 
6.6883 

743*4 

.6691 

I-3383 
2.0074 
2.6765 
3-3457 
4.0148 
4.6839 
5-353° 

6.0222 

6.6913" 

~T 

2 
3 
4 
5 

6 

7 
8 
9 
10 

48f  Deg. 

48  J  Deg. 

48  i  Deg. 

48  Deg. 

42|  Deg. 

42  £  Deg. 

42|  Deg. 

43  Deg. 

.7402 
1.4804 
2.2207 
2.9609 
3.7011 

4.4413 
5-1815 
5-9217 
6.6620 
7.4022 

.6724 

1-3447 
2.0171 
2.6895 
3.3618 

4.0342 
4.7066 

5-3789 
6.0513 

6.7237 

•7373 
1.4746 

2.  2Il8 
2.9491 
3.6864 

4-4237 
5.1609 
5.8982 

6-6355 

7.3728 

.6756 
1.3512 
2.0268 

2.7024 
3-378o 

4-°535 
4.7291 

5-4047 
6.0803 

6-7559 

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2.2030 
2.9373 
3.6716 

4.4059 

5'£4°| 

5.8746 
6.6089 
7-3432 

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i'3576 
2.0364 
2.7152 
3-3940 
4.0728 
4.7516 

5-4304 
6.1092 
6.7880 

•7|i.4 

1.4627 
2.1941 
2.9254 
3.6568 

4.3881 

5-"95 
5-8508 
6.5822 

7-3J35 

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1.3640 
2.0460 
2.7280 

3.4100 

4.0920 

4.7740 
5.4560 
6.1380 
6.8200 

1 
2 
3 
4 
5 

6 

7 
8 
9 
1O 

1 
2 
3 
4 
5 

6 

2 

9 
10 

471  Deg. 

47i  Deg. 

47i  Deg. 

47  Deg. 
44  Deg. 

43  J  Deg. 

43  J  Deg. 

43f  Deg. 

.7284 
1.4567 
2.1851 

2-9135 
3.6419 

4.3702 
5.0986 
5.8270 

6-5553 

7.2837 

.6852 

I-3704 
2.0555 
2.7407 
3-4259 
4.11  ii 
4.7963 

5-4^5 
6.1666 
6.8518 

•7254 
1.4507 
2.1761 
2.901.5 
3.6269 

4.3522 
5.0776 
5.8030 
6.5284 

7-2537 

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1.3767 
2.0651 

2-7534 
3.4418 

4.1301 
4.8185 
5.5068 
6.1952 
6-8835 

Lat. 

.7224 
1.4447 
2.1671 
2.8895 
3.6118 

4-3342 
5-0565 
5-7789 
6.5013 
7.2236 

.6915 

1.3830 
2.0745 
2.7661 
3-4576 

4.1491 
4.8406 
5-5321 
6.2236 
6.9151 

.7193 

1-4387 
2.1580 
2.8774 
3-5967 
4.3160 
5-0354 
5-7547 
6.4741 

7-1934 

•6947 
1-3893 

2.0840 

2.7786 

3-4733 
4.1680 
4.8626 

5-5573 
6.2519 
6.9466 

1 
2 
3 
4 
5 

6 

7 
8 
9 
10 

1 

D. 

Dep. 

Lat. 

Dep. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

46f  Deg. 

46*  Deg. 

46J  Deg. 

46  Deg. 

15 


3 

3  AND  DEPARTURES. 

jATITUDE, 

D. 

44J  Deg. 

44J  Deg. 

44f  Deg. 

45  Deg. 

D. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

.7163 

.6978 

•7133 

.7009 

.7102 

.7040 

.7071 

.7071 

1 

2 

1.4326 

1.3956 

1.4265 

1.4018 

1.4204 

1.4080 

1.4142 

1.4142 

2 

1    3 
4 

2.1489 
2.8652 

2.0934 
2.7912 

2.1398 
2.8530 

2.1027 
2.8036 

2.1306 

2.8407 

2.II20 

2.8161 

2.1213 

2.8284 

2.1213 

2.8284 

3 
4 

5 

3-58I5 

3.4890 

3.5663 

3-5°45 

3-55°9 

3.5201 

3-5355 

3-5355 

5 

6 

4.2978 

4.1867 

4.2795 

4.2055 

4.2611 

4.2241 

4.2426 

4.2426 

6 

7 

5.0141 

4.8845 

4.9928 

4.9064 

4-97!3 

4.9281 

4-9497 

4-9497 

7 

8 
9 

5-7304 
6.4467 

5-5823 
6.2801 

5.7060 
6.4193 

5.6073 
6.3082 

5.6815 
6.3917 

5.6321 
6.3361 

5.6569 
6.3640 

5.6569 
6.3640 

«  1 
9 

1O 

7.1630 

6.9779 

7.1325 

7.0091 

7.1019 

7.0401 

7.0711 

7.0711 

10 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

D. 

D. 

45f  Deg. 

45  *  Deg. 

451  Deg. 

45  Deg. 

TABLE  OF  USEFUL  NUMBERS. 

Logarithms. 
Ratio  of  circumference  to  diameter  n  =  3.1415916536 0.4971499 

Area  of  circle  to  radius  i  = "  " 

Surface  of  sphere  to  diameter  i  = " " 

Area  of  circle  to  diameter  i  =  7853981634 — 1.8950899 

Base  of  Napierian  Logarithms  = 2.7182818285 4342945 

Modulus  of  common        "          = 4342944819 — 1.6377843 

Equatorial  radius  of  the  earth,  in  feet  =  20923599.98 7.3206364 

Polar  «  "  «     =  20853657.16 7.3191823 

Length  of  seconds  pendulum,  in  London,  in  inches  =  39.13929. 

"  "  "  Paris  «      =  39.1285. 

"  "  "  New  York       «      =39.1012. 

U.  S.  standard  gallon  contains  231  c.  in.,  or  58372.175  grains  =  8.338882  Ibs.  avoir- 
dupois of  water  at  39.8°  Fahr. 
U.  S.  standard-bushel  contains  2150.42  c.  in.,  or  77.627413  Ibs.  av.  of  water  at  39.8° 

Fahr. 

British  imperial  gallon  contains  277.274  c.  in.,  =  1.2003  wine  gallons  of  231  c.  in. 
French  metre  =  39.37079  in.  =  3.28089917  feet. 

"       toise  =  6.39459252  feet. 

u       are  =  100  sq.  metres  =  1076.4299  sq.  ft. 

"       hectare  =  100  ares  =  2.471143  acres  =  107642.9936  sq.  ft. 

"       litre  =  I  cubic  decimeter  =  61.02705  c.  in.  =.26418637  gallons  of  231  c.  in. 

"       hectolitre  =  100  litres  =  26.418637  gallons. 
1  pound  avoirdupois  =  7000  grs.  =  1.215277  pounds  Troy. 
I      "      Troy  =  5760  grs.  =  .822857  pounds  avoir. 
i  gramme  =  15.442  grains. 

i  kilogramme  =  1000  grammes  =  15442  grs.  =  2.20607  tt>s.  avoir. 
Tropical  year  =  365  d.  5  h.  45  m.  47.588  sec. 


1G 


TABLE 


OF   THE 


LOGARITHMS    OF  NUMBERS, 


FROM 


1   TO    10,000. 


ir 


A  TABLE 


OS  THE 


LOGARITHMS  OF  NUMBERS 


FKOM   1   TO   10,000. 


N. 

Log. 

N. 

Log. 

N. 

Ug. 

N. 

Log. 

1 

o.oooooo 

26 

1.414973 

51 

1.707570 

76 

1.880814 

2 
3 

0.301030 
0.477121 
0.602060 

27 
28 
29 

1.431364 
1.447158 
1.462398 

52 
53 
54 

1.716003 

1.724276 
1.732394 

17 
78 
79 

1.886491 
1.892095 
1.897627 

5 

0.698970 

30 

1.477121 

55 

1.740363 

80 

1.903090 

6 

0.778151 

31 

1.491362 

56 

1.748188 

81 

1.908485 

7 

0.845098 

32 

1.505150 

57 

1-755875 

82 

1.913814 

8 

0.903090 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

o-954243 

34 

i-53I479 

59 

1.770852 

84 

1.924279 

10 

1.  000000 

35 

1.544068 

60 

1.778151 

85 

1.929419 

11 

1-041393 

36 

i-5563°3 

61 

1-78533° 

86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

13 

•IJ3943 

38 

1.579784 

63 

I-799341 

88 

1.944483 

14 
15 

.146128 
.176091 

39 
40 

1.591065 
1.602060 

64 
65 

i.  806180 
1.812913 

89 
90 

1.949390 
I-954H3 

16 

.204120 

41 

1.612784 

66 

1.819544 

91 

1.959041 

17 
18 

.230449 

.255273 

42 
43 

1.623249 
1.633468 

67 
68 

1.826075 
1.832509 

92 
93 

1.963788 
1.968483 

19 

.278754 

44 

I-643453 

69 

1.838849 

94 

1.973128 

20 

.301030 

45 

1.653213 

70 

1.845098 

95 

1.977724 

21 

.322219 

46 

1.662758 

71 

1.851258 

96 

1.982271 

22 

.342423 

47 

1.672098 

72 

1.857332 

97 

1.986772 

23 

.361728 

48 

1.681241 

73 

i  863323 

98 

1.991226 

24 

.380211 

49 

1.690196 

74 

i  869232 

99 

1.995635 

25 

.397940 

50 

1.698970 

75 

1.875061 

100 

2.OOOOOO 

19 


N.  100.          LOGARITHMS.       Log.  000. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

oooooo 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891  1 

101 

^321 

4751 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

102 

103 

600 

012837 

9026 
3259 

945i 
3680 

9876 
4100 

°3oo 
4521 

0724 
4940 

'147 
5360 

'570 
5779 

J993 
6197 

84'5 
6616 

104 

7033 

745  1 

7868 

8284 

8700 

9116 

9532 

9947 

°36i 

°775 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4°75 

4486 

4896 

106 
107 

5306 
9384 

57J5 
9789 

6125 
°i95 

6533 
°6oo 

6942 
'004 

7350 
'408 

7757 

>8l2 

8164 

221  6 

8571 
2619 

8978 

302I 

108 
109 

033424 
7426 

3826 
7825 

4227 
8223 

4628 
8620 

5029 
9017 

543° 
9414 

5830 
9811 

6230 
°207 

6629 

%02 

7028 

110 
111 

041393 

5323 

1787 
57H 

2182 
6105 

2576 

6n425 

2969 

6885 

3362 
7275 

3755 
7664 

4148 

81°23 

4540 
8442 

4932 
8830 

112 

9218 

9606 

9993 

0380 

0766 

J53 

J538 

2309 

2694 

113 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

0320 

115 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

116 

4458 

4832 

5206 

558o 

5953 

6326 

6699 

7071 

7443 

7815 

117 

8186 

8557 

8928 

9298 

9668 

0038 

0407 

0776 

•145 

'514 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

445  * 

4816 

5182 

119 

5547 

5912 

6276 

6040 

7004 

7368 

7731 

8094 

8457 

8819 

120 

079181 

9543 

9904 

°266 

0626 

0987 

'347 

1707 

2067 

2426 

121 

082785 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

122 

6360 

6716 

7071 

7426 

8136 

8490 

8^45 

9198 

9552 

123 

99°5 

0258 

°6n 

0963 

?V5 

'667 

2018 

272I 

3o7i 

124 

093422 

3772 

4122 

4820 

5169 

55l8 

5866 

6215 

6562 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

0026 

126 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

5l69 

5510 

5851 

6191 

6531 

687i 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

0253 

129 

110590 

0926 

1263 

'599 

1934 

2270 

2605 

2940 

3^75 

3609 

130 

"3943 

4277 

4611 

4944 

5^78 

5611 

5943 

6276 

6608 

6940 

131 

7271 

7603 

7934 

8265 

8926 

9256 

9586 

99*5 

0245 

132 

120574 

0903 

1231 

1560 

_  0  O  O 

I  OOO 

2216 

2544 

2871 

3198 

3525 

133 

385* 

4178 

45°4 

4830 

5156 

5481 

5806 

6131 

6456 

678i 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

°OI2 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

136 

3539 

3858 

4177 

4496 

4814 

5i33 

5451 

5769 

6086 

6403 

137 

6721 

7°37 

7354 

7671 

7987 

8303 

8618 

°934 

9249 

9564 

138 

9879 

°i94 

°5o8 

°822 

1136 

'450 

*7^3 

2076 

2389 

2702 

139 

143015 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

5818 

140 

146128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

141 

9219 

9527 

9835 

°I42 

°449 

o756 

1370 

'676 

1982 

142 

152288 

2594 

2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

143 

5336 

5640 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8o6l 

144 

8362 

8664 

8965 

9266 

9567 

9868 

°i68 

0469 

0769 

'068 

145 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

146 

4353 

4650 

4947 

5244 

5541 

5838 

6134 

6430 

6726 

7O22 

147 

73*7 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

148 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

149 

3186 

3478 

3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

150 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

151 

8977 

9264 

9552 

9839 

°I26 

0413 

0699 

0986 

1272 

1558 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4I23 

4407 

153 

4691 

4975 

5259 

5542 

5^5 

6108 

6391 

6674 

6956 

7239 

154 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

977i 

0051 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

156 

3403 

3681 

3959 

4237 

45H 

4792 

5069 

5346 

5623 

157 
158 

5900 
8657 

8932 

6453 
9206 

6729 
9481 

7005 
9755 

7281 

7556 
0303 

7832 
°577 

8io7 
0850 

8382 
'124 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3°33 

33°5 

3577 

3848 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

20 


N.  160.         LOGARITHMS.       Log.  204.  ; 

N. 

O 

i 

2 

3 

4 

5 

6 

7 

8 

9   | 

160 

204120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556  i 

161 

6826 

7096 

7365 

7634 

7904 

8441 

8710 

8979 

9*47 

162 

9515 

9783 

0051 

0586 

°853 

1388 

'654 

'921 

163 

212188 

2454 

2720 

2986 

3*5* 

35i8 

3783 

4049 

43J4 

4579 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

165 

7484 

7747 

8010 

8*73 

8536 

8798 

9060 

93*3 

9585 

9846  j 

166 

220108 

0370 

0631 

0892 

"53 

1414 

1675 

1936 

2196 

2456 

167 

2716 

2976 

3*36 

3496 

3755 

4015 

4*74 

4533 

479* 

5051 

168 

53°9 

5568 

5826 

6084 

6342 

6600 

6858 

7"5 

737* 

7630 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

o,93 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

171 

2996 

3250 

35°4 

3757 

4011 

4264 

45*7 

4770 

5°*3 

5276 

172 

55*8 

6033 

6285 

6537 

6789 

7041 

7292 

7Q544 

7795 

173 

8046 

8*97 

8548 

8799 

9049 

9*99 

955° 

9800 

0300 

174 

240549 

0799 

1048 

i*97 

I546 

1795 

2044 

2293 

2541 

2790 

175 

3038 

3*86 

3534 

3782 

4030 

4*77 

45*5 

477* 

5019 

5266 

176 

55*3 

5759 

6006 

6252 

6499 

6745 

6991 

7*37 

7482 

77*8 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

993* 

0176 

178 
179 

250420 
*853 

0664 
3096 

0908 
3338 

1151 

1395 

3822 

1638 
4064 

1881 
4306 

2125 
4548 

2368 
4790 

2610 
5031 

180 

255273 

55H 

5755 

~5996 

6237 

6477 

6718 

6958 

7198 

7439 

181 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

182 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

184 

4818 

5°54 

5290 

55*5 

5761 

5996 

6232 

6467 

6702 

6937 

185 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9*79 

186 

95i3 

9746 

9980 

°*i3 

°44^ 

°679 

°9I2 

!>44 

,  '377 

!6o9 

187 

271842 

2074 

2306 

*538 

2770 

3001 

3*33 

3464 

3696 

39*7 

188 
189 

f£8 
6462 

6692 

4620 
0921 

4850 
7151 

5081 
7380 

53" 
7609 

554* 
7838 

577* 
8067 

6002 
8296 

6232 
85*5 

190 

278754 

8982 

9211 

9439 

9667 

9895 

°I23 

°35' 

°578 

°8o6 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3°75 

192 
193 

3301 

5557 

35*7 
5782 

3753 
6007 

3979 
6232 

4205 
6456 

443  1 
6681 

4656 
6905 

4882 
7130 

5107 
7354 

533* 
7578 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9'43 

9366 

9589 

9812 

195 

290035 

0257 

0480 

0702 

0925 

"47 

1369 

1591 

1813 

2034 

196 

2256 

2478 

2699 

2920 

3363 

3584 

3804 

4025 

4246 

197 

rr 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

198 

6665 

6884 

7104 

73*3 

754* 

7761 

7979 

8198 

8416 

8635 

199 

8853 

9071 

9289 

9507 

97*5 

9943 

°i6i 

o378 

°595 

°8i3 

200 

301030 

i*47 

1464 

1681 

1898 

2114 

2331 

*547 

2764 

2980 

201 

3196 

3628 

3844 

4059 

4*75 

449  1 

4706 

49*  i 

5136 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

203 

7496 

7710 

79*4 

8i37 

8351 

8564 

8778 

8991 

9204 

9417 

204 

9630 

9843 

0056 

°268 

°48i 

0693 

0906 

»n8 

1330 

'54* 

205 

3"754 

1966 

2177 

2389 

2600 

2812 

3023 

3*34 

3445 

3656 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5*3° 

534° 

5551 

5760 

207 

597° 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

208 

8063 

8272 

8481 

8689 

8898 

9106 

93*4 

95** 

9730 

9938 

209 

320146 

0354 

0562 

0769 

0977 

1184 

I391 

1598 

1805 

2OI2 

210 

322219 

2426 

T633 

2839 

3046 

~3*5* 

3458 

3665 

3871 

4077 

211 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

57*i 

5926 

6131 

212 

6336 

6541 

6745 

6950 

7*55 

7359 

7563 

7767 

797* 

8176 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

°oo8 

°2II 

214 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

215 

2438 

2640 

2842 

3°44 

3246 

3447 

3649 

3850 

4051 

i2!3 

216 

4454 

4655 

4856 

5°57 

5*57 

5458 

5^58 

5859 

6059 

6260 

217 

T*3~ 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8*57 

218 

8456 

8656 

8855 

9054 

9*53 

945  i 

9650 

9849 

o047 

°246 

219 

340444 

0642 

0841 

1039 

i*37 

H35 

1632 

1830 

2028 

2225 

!  N. 

0 

1 

2 

3 

* 

5 

6 

7 

8 

9 

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945 

5432 

5478 

55*4 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

948 

6808 

6854 

6900 

6946 

6992 

7°37 

7083 

7129 

7175 

7220 

949 

7266 

7312 

7358 

74°3 

7449 

7495 

7541 

7586 

7632 

7678 

950 

977724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8i35 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

954 

9548 

9594 

9639 

9685 

973° 

9776 

9821 

9867 

9912 

9958 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

958 

1366 

lftl 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

J773 

959 

1819 

1864 

1909 

'954 

2000 

2045 

2090 

2135 

2181 

2226 

~960" 

982271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

962 

3*75 

3220 

3265 

3310 

3356 

3401 

3446 

349i 

3536 

358i 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

964 

4077 

4122 

4167 

4212 

4^57 

4302 

4347 

4392 

4437 

4482 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

4977 

5022 

5067 

5112 

5'57 

5202 

5*47 

5292 

5337 

5382 

967 

5426 

547i 

55i6 

556i 

5606 

5651 

5696 

5741 

5786 

583° 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

986772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

971 

7219 

7264 

73°9 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

77" 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

973 

8"3 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

85H 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

975 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

976 
977 

9450 
9895 

9494 
9939 

9539 
9983 

9583 

°028 

9628 

°072 

9672 
°U7 

9717 
°i6i 

9761 
0206 

9806 

0250 

9850 
°*94 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

"37 

1182 

980 

991226 

1270 

1315 

1359 

1403 

1448 

1492 

I536 

1580 

1625 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

982 

2III 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

983 

*554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

984 

2995 

3°39 

3083 

3127 

3172 

3216 

3260 

33°4 

3348 

3392 

985 

3436 

3480 

3524 

3568 

36l3 

3657 

3701 

3745 

3789 

3833 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

43r7 

4361 

44°5 

4449 

4493 

4537 

458i 

4625 

4669 

4713 

988 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

989 

5J96 

5240 

5284 

53^8 

5372 

5416 

5460 

55°4 

5547 

5591 

990 

995635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7°37 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

743° 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

997 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

998 

9131 

9*74 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

!  N« 

0 

1 

2 

3 

4: 

5 

~6~ 

7 

8 

9 

TABLE 


OF 


LOGARITHMIC    SINES 


TANGENTS. 


35 


0°            £OCU 

[C           179° 

LRXTH1MU 

M. 

Sec. 

Sine. 

Tang. 

M. 

Sec. 

Sine. 

Tang. 

0 

~60 

10 

7.463725 

7.463727 

50  I 

10 
20 
30 

5-685575 
5.986605 
6.162696 

5.685575 
6.162696 

50 
40 
30 

10 

20 
30 

70904 
77966 
_  84915 

70906 
77968 
84917 

50 
40 
30 

40 
50 

.287635 
.384545 

.287635 
•384545 

20 
10 

40 
50 

QI7<4 
735*4*7 

7-598490 

20 
10 

1 

10 

•4637*6 
.530673 

.463726 
•530673 

50 

59 

11 

10 

7.505118 
11649 

05120 
11651 

50 

49 

20 

.588665 

.588665 

40 

20 

18083 

18085 

40 

30 

.619817 

.639817 

30 

30 

24423 

24426 

30 

40 

•685575 

•685575 

20 

40 

30672 

30675 

20 

50 

.726968 

.726968 

10 

50 

36832 

36835 

10 

2 

.764756 

•764756 

~58 

12 

42906 

42909 

48 

10 

.799518 

.799518 

50 

10 

48897 

48899 

50 

20 

.831703 

.831703 

40 

20 

54806 

54808 

40 

30 

.861666 

.861666 

30 

30 

60635 

60638 

30 

40 

.889695 

.889695 

20 

40 

66387 

66390 

20 

50 

.916024 

.916024 

10 

50 

72065 

72068 

10 

3 

.940847 

.940847 

57 

13 

77668 

77671 

47 

10 

.964328 

.964329 

50 

10 

83201 

83204 

50 

20 

6.986605 

6.986605 

40 

20 

88664 

88667 

40 

30 

7.007794 

7.077794 

30 

30 

94°59 

94062 

30 

40 
50 

27998 
47303 

27998 
473°3 

20 
10 

40 
50 

7-599388 
7.604652 

7-599391 
7.604655 

20 
10 

4 

65786 

65786 

56 

14 

09853 

09857 

46 

10 

7.08351? 

7.083515 

50 

10 

H993 

14996 

50 

20 

7.100548 

7.100548 

40 

20 

20072 

20076 

40 

30 

16938 

16939 

30 

30 

25091 

25097 

30 

40 

3*733 

3*733 

20 

40 

^co5§ 

30060 

20 

50 

47973 

47973 

10 

50 

34962 

34968 

10 

5 

1*696 

62696 

55 

15 

Kill 

%  C  ?20 

45 

10 

76936 

76937 

50 

10 

44615 

446*9 

50 

20 

7.190725 

7.190725 

40 

20 

49361 

49366 

40 

30 

7.204089 

7.204089 

30 

30 

54056 

54061 

30 

40 

17054 

17054 

20 

40 

58701 

58706 

20 

50 

29643 

29643 

10 

50 

63297 

63301 

10 

6 

41877 

41878 

~54 

16 

67845 

67849 

44 

10 

53776 

53777 

50 

10 

7*345 

72350 

50 

20 
30 

65358 
76639 

6I159 
76640 

40 
30 

20 
30 

76799 
81208 

76804 
8l2I1 

40 
30 

40 
50 

87635 
7.298358 

87635 
7.298359 

20 
10 

40 
60 

85573 
89894 

8557§ 
89900 

20 
10 

7 

7.308824 

7.308825 

53 

17 

94173 

94179 

43 

10 

19043 

19044 

50 

10 

7.698410 

7.698416 

50 

20 

29027 

29028 

40 

20 

7.702606 

7.7026l2 

40 

30 

38787 

38788 

30 

30 

06762 

06768 

30 

40 

48332 

48333 

20 

40 

10879 

10885 

20 

50 

57672 

57673 

10 

50 

14957 

14962 

10 

8 

66816 

66817 

52 

18 

18997 

19003 

42 

10 

75770 

7577* 

50 

10 

22999 

23005 

50 

20 

84544 

84546 

40 

20 

26966 

26972 

40 

30 

7-393H5 

7-393H6 

30 

30 

30896 

30902 

30 

40 

7.401578 

7-4OI579 

20 

40 

34791 

34797 

20 

50 

09850 

09852 

10 

50 

38651 

38658 

10 

9 

17968 

17970 

51 

19 

4*477 

42484 

41 

10 

25937 

25939 

50 

10 

46270 

46277 

50 

20 

33762 

33764 

40 

20 

50031 

50037 

40 

30 

41449 

41451 

30 

30 

53758 

53765 

30 

40 

49002 

49004 

20 

40 

57454 

57462 

20 

50 

56426 

56428 

10 

50 

61119 

61127 

10 

10 

7.463725 

7.463727 

50 

20 

7.764754 

7.764761 

4O 

Cosine. 

Cotang. 

Sec 

M. 

Cosine. 

Cotang. 

s^cT 

M. 

90°                                       89° 

36 


0°                  SINES  AND  TANGENTS.                179° 

M. 

Sec. 

Sine. 

Tang. 

M. 

Sec. 

Sine. 

Tang. 

20 

7.764754 

7.764761 

40 

30 

7.940842 

7.940858 

30 

10 

68365 

50 

10 

43*48 

43265 

50 

20 

71932 

71940 

40 

20 

45641 

45657 

40 

30 

75477 

75485 

30 

30 

48020 

48037 

30 

40 

78994 

79002 

20 

40 

50387 

50404 

20 

50 

82482 

82490 

10 

50 

52741 

5*758 

10 

21 

85943 

85951 

39 

31 

55082 

55100 

29 

10 

89376 

893H 

50 

10 

574*0 

574*8 

50 

20 
30 

92782 
96162 

92790 
96170 

40 
30 

20 
30 

597*7 
62031 

59745 
62049 

40 
30 

40 
50 

7-799515 
7-80^843 

7-7995*4 
7.802852 

20 
10 

40 
50 

66602 

66621 

20 
10 

22 

06146 

06155 

~38 

32 

68870 

68889 

28 

10 
20 

09423 
12677 

09472 
12686 

50 
40 

10 

20 

71126 
73370 

7**45 
73389 

50 
40 

30 

J59O5 

15915 

30 

30 

75603 

75622 

30 

40 

19111 

19120 

20 

40 

77824 

77844 

20 

50 

22292 

22302 

10 

50 

80034 

80054 

10 

23 

*545* 

25460 

37 

33 

82233 

82253 

27 

10 

28586 

28596 

50 

10 

84421 

84441 

50 

20 

31700 

31710 

40 

20 

86598 

86618 

40 

30 

34791 

348oi 

30 

30 

88764 

88785 

30 

40 

37860 

37870 

20 

40 

90919 

90940 

20 

50 

40907 

40918 

10 

50 

93064 

93085 

10 

"24 

10 

43934 
46939 

43944 
46950 

50 

~36 

~34 

10 

95198 
97322 

95219 

97343 

50 

26 

20 

499*4 

49935 

40 

20 

7-999435 

7.999456 

40 

30 

52888 

52900 

30 

30 

8.001538 

8.001560 

30 

40 

55833 

55844 

20 

40 

03631 

03653 

20 

50 

58757 

58769 

10 

50 

05714 

05736 

10 

25 

61662 

61674 

35 

35 

07787 

07809 

25 

10 

20 

64548 
67414 

64560 
67426 

50 

40 

10 

20 

09850 
11903 

09872 
11926 

50 

40 

30 

70262 

70274 

30 

30 

13947 

13970 

30 

40 

73092 

73104 

20 

40 

15981 

16004 

20 

50 

75902 

759*5 

10 

50 

18005 

18029 

10 

~26 

78695 

78708 

in 

~36 

20021 

20044 

24 

10 

81470 

81483 

50 

10 

22027 

22051 

50 

20 

84228 

84240 

40 

20 

24023 

24047 

40 

30 

86968 

86981 

30 

30 

260II 

26035 

30 

40 

89690 

89704 

20 

40 

27989 

28014 

20 

50 

92396 

92410 

10 

50 

29959 

29984 

10 

27 

95085 

95099 

33 

37 

3I9I9 

3*945 

23 

10 

7-897758 

7.897771 

50 

10 

33»7I 

33897 

50 

20 

7.900414 

7.900428 

40 

20 

35840 

40 

30 
40 
50 

03054 
05678 
08287 

03068 
05692 
08301 

30 
20 
10 

30 
40 
50 

37749 
39675 
41592 

37775 
39701 
41618 

30 
20 
10 

28 

10879 

10894 

~32 

38 

43501 

435*7 

22~ 

10 

13457 

13471 

50 

10 

45401 

454*8 

50 

20 

16019 

16034 

40 

20 

47*94 

40 

30 

18566 

18581 

30 

30 

49178 

49*05 

30 

40 

21098 

21113 

20 

40 

5*054 

51081 

20 

50 

23616 

23631 

10 

50 

52922 

5*949 

10 

29 

26119 

26134 

31 

39 

54781 

54809 

/?! 

10 

28608 

2862! 

50 

10 

56633 

56661 

50 

20 
30 

31082 
33543 

31098 
33559 

40 
30 

20 
30 

58477 
60314 

58506 
60342 

40 
30 

40 
50 

35989 
38422 

36006 
38439 

20 
10 

40 
50 

62142 
63963 

62171 
63992 

20 
10 

30 

7.940842 

7.940858 

30 

40 

8.065776 

8.065806 

20 

Cosine. 

Cotang. 

Sec. 

IT 

Cosine. 

Cotang. 

SecT 

M. 

90°                                                                                               89° 

24 


37 


0°             LOGARITHMIC           179° 

M. 

Sec. 

Sine. 

Tang. 

M. 

Sec. 

Sine. 

Tang. 

40 

8.065776 

8.065806 

20 

50 

8.162681 

8.162727 

10 

10 

67582 

67612 

50 

10 

64126 

64172 

50 

20 

69380 

69410 

40 

20 

65566 

65613 

40 

30 

71171 

71201 

30 

30 

67002 

67049 

30 

40 

7*955 

72985 

20 

40 

68433 

68480 

20 

50 

74731 

74761 

10 

50 

69859 

69906 

10 

41 

76500 

76531 

19 

51 

71280 

71328 

9 

10 

78261 

78293 

50 

10 

72697 

72745 

50 

20 

80016 

80047 

40 

20 

74109 

74158 

40 

30 

81764 

81795 

30 

30 

755*7 

75566 

30 

40 

83504 

83536 

20 

40 

76920 

76969 

20 

50 

'85238 

85270 

10 

50 

78319 

78368 

10 

~4Z 

86965 

86997 

~18 

~52 

79713 

79763 

8 

10 

88684 

88717 

50 

10 

81102 

81152 

50 

20 

90398 

90430 

40 

20 

82488 

82538 

40 

30 
40 
50 

92104 
93804 
95497 

92137 
93837 

9553° 

30 
20 
10 

30 
40 
50 

83868 

85245 
86617 

83919 
•85296 
86668 

30 
20 
10 

43 

10 

20 

8.098863 
8.100537 

97217 
8.098897 
8.100571 

50 

40 

17 

53 

10 
20 

87985 
89348 
90707 

88036 
89400 
90760 

50 
40 

7 

30 

02204 

02239 

30 

30 

92062 

92115 

30 

40 

03864 

03899 

20 

40 

934*3 

93466 

20 

50 

05519 

°5554 

10 

50 

94760 

94813 

10 

44 

10 

07167 
08809 

07202 
08845 

50 

^16 

54 

10 

96102 

97440 

96156 
97494 

50 

6 

20 

10444 

10481 

40 

20 

8.198774 

8.198829 

40 

30 

12074 

IZIIO 

30 

30 

8.200104 

8.200159 

30 

40 

13697 

13734 

20 

40 

01430 

01485 

20 

50 

'53*5 

15352 

10 

50 

02752 

02808 

10 

45 

16926 

16963 

15 

55 

04070 

04126 

5 

10 

20 

18532 
20131 

18569 
20169 

50 

40 

10 

20 

05384 
06694 

05440 
0.6750 

50 

40 

30 

21725 

21763 

30 

30 

08000 

08057 

30 

40 

23313 

23351 

20 

40 

09302 

09359 

20 

50 

24895 

24933 

10 

50 

10601 

10658 

10 

~46 

8.126471 

8.126510 

~14 

56 

11895 

JI953 

4 

10 

28042 

28081 

50 

10 

13185 

J3243 

50 

20 

29606 

29646 

40 

20 

14472 

14530 

40 

30 

31166 

31206 

30 

30 

J5755 

15814 

30 

40 
50 

32720 
34268 

32760 
34308 

20 
10 

40 
50 

17034 
18309 

17093 
18369 

20 
10 

47 

35810 

35851 

13 

57 

8.219581 

8.219641 

3 

10 

37348 

37389 

50 

10 

20849 

20909 

50 

20 

38879 

38921 

40 

20 

22113 

22174 

40 

30 

40406 

40447 

30 

30 

23374 

23434 

30 

40 

41927 

41969 

20 

40 

24631 

24692 

20 

50 

43443 

43485 

10 

50 

25884 

25945 

10 

~48 

10 

44953 
46458 

44996 
46501 

50 

~12 

58 

10 

27133 
28380 

27195 
28442 

50 

~2~ 

20 

47959 

48001 

40 

20 

29622 

29685 

40 

30 

49453 

49497 

30 

30 

30861 

30924 

30 

40 

50943 

50987 

20 

40 

32096 

32160 

20 

50 

52428 

52472 

10 

50 

33328 

33392 

10 

49 

53907 

53952 

11 

59 

34557 

34621 

1 

10 

553^ 

55426 

50 

10 

357»2 

35846 

50 

20 

56852 

56896 

40 

20 

37003 

37068 

40 

30 

58316 

58361 

30 

30 

38221 

38286 

30 

40 
50 

59776 
61231 

59821 
61276 

20 
10 

40 
50 

39436 
40647 

395oi 
407i3 

20 
10 

50 

8.162681 

8.162727 

10 

60 

8.241855 

8.241921 

0 

Cosine. 

Cotang. 

Sec. 

M. 

Cosine. 

Cotang. 

Sel" 

M. 

90°                                         89° 

88 


1°        SINES  AND  TANaENTS.       178° 

M. 

Sec. 

Sine. 

Tang. 

M. 

Sec. 

Sine. 

Tang. 

O 

8.241855 

8.241921 

60 

10 

8.308794 

8.308884 

50 

10 

3060 

3126 

50 

10 

8.309827 

8.309917 

50 

20 

4261 

43*8 

40 

20 

8.310857 

8.310948 

40 

30 

5459 

55*6 

30 

30 

1885 

1976 

30 

40 

6654 

6721 

20 

40 

2910 

3002 

20 

50 

7845 

7913 

10 

50 

3933 

4025 

10 

1 

8.249033 

8.249101 

59 

11 

4954 

5046 

49 

10 

8.250218 

8.250287 

50 

10 

597* 

6065 

50 

20 

1400 

1469 

40 

20 

6987 

7081 

40 

30 

2578 

2648 

30 

30 

8001 

8095 

30 

40 

3753 

38*3 

20 

40 

8.319012 

8.319106 

20 

50 

49*5 

4996 

10 

50 

8.320021 

8.320115 

10 

2 

6094 

6165 

58 

12 

1027 

1122 

48 

10 

7260 

7331 

50 

10 

2031 

2127 

50 

20 

8423 

8494 

40 

20 

3033 

3129 

40 

30 

8.259582 

8.259654 

30 

30 

4032 

4128 

30 

40 

8.260739 

8.260811 

20 

40 

5029 

5126 

20 

50 

1892 

1965 

10 

50 

6024 

6121 

10 

3 

3042 

3JI5 

57 

13 

7016 

7114 

47 

10 

4190 

4263 

50 

10 

8007 

8105 

50 

20 

5334 

5408 

40 

20 

8995 

8.329093 

40 

30 
40 

6475 
7613 

ps 

30 
20 

30 
40 

8.329980 
8.330964 

8.330080 
1064 

30 
20 

50 

8749 

8824 

10 

50 

1945 

2045 

10 

* 

10 

8.269881 
8.271010 

8.269956 
8.271086 

50 

~56 

14 

10 

2924 
3901 

3025 
4002 

50 

46 

20 

2137 

2213 

40 

20 

4876 

4977 

40 

30 

3260 

3337 

30 

30 

5848 

595° 

30 

40 

4381 

4458 

20 

40 

6819 

6921 

20 

50 

5499 

5576 

10 

50 

7787 

7890 

10 

5 

6614 

6691 

55 

15 

8753 

8856 

45 

10 

77*6 

7804 

50 

10 

8.339717 

8.339821 

50 

20 

8835 

8.278913 

40 

20 

8.340679 

8.340783 

40 

30 

8.279941 

8.280020 

30 

30 

1638 

1743 

30 

40 

8.281045 

1124 

20 

40 

2596 

2701 

20 

50 

"45 

2225 

10 

50 

3657 

10 

6 

3*43 

33*3 

~54 

16 

45  °4 

4610 

44~ 

10 

4339 

4419 

50 

10 

5456 

5562 

50 

20 

543  * 

40 

20 

6405 

6512 

40 

30 
40 

6521 

7608 

6602 
7689 

30 
20 

30 
40 

•  735* 
8*97 

7459 
8405 

30 
20 

50 

8692 

8774 

10 

50 

8.349240 

8.349348 

10 

7 

8.289773 

8.289856 

53 

17 

8.350181 

8.350289 

43 

10 

8.290852 

8.290935 

50 

10 

1119 

1229 

50 

20 

1928 

2O  I  2 

40 

20 

2056 

2166 

40 

30 

3002 

3086 

30 

30 

2991 

3101 

30 

40 

4073 

4157 

20 

40 

39*4 

4°35 

20 

50 

5141 

5226 

10 

50 

4855 

4966 

10 

8 

6207 

6292 

~52 

18 

5783 

5f95 

42 

10 

7270 

7355 

50 

10 

6710 

6823 

50 

20 

8330 

8416 

40 

20 

7635 

7748 

40 

30 

8.299388 

8.299474 

30 

30 

8558 

8671 

30 

i 

40 

8.300443 

8.300530 

20 

40 

8-359479 

8-359593 

20 

50 

1496 

1583 

10 

50 

8.360398 

8.360512 

10 

9 

2546 

2633 

51 

19 

1315 

1430 

41 

10 

3594 

3682 

50 

10 

2230 

*345 

50 

20 

4639 

47*7 

40 

20 

3H3 

3*59 

40 

30 
40 

5681 
6721 

5770 
6811 

30 
20 

30 
40 

4054 
4964 

4171 
5080 

30 
20 

50 

7759 

7849 

10 

50 

5871 

5988 

10 

10 

8.308794 

8.308884 

50 

20 

8.366777 

8.366894 

40 

Cosine. 

Cotang.  j  Sec. 

M. 

Cosine.    Cotang. 

SecT 

M. 

91°                                       88° 

39 


1°             LOGARITHMIC           178° 

M. 

Sec. 

Sine. 

Tang. 

M. 

Sec. 

Sine. 

Tang. 

20 

8.366777 

8.366894 

40 

~30 

8.417919 

8.418068 

30 

10 

7681 

7799 

50 

10 

8722 

8872 

50 

20 

8582 

8701 

40 

20 

8.419524 

8.419674 

40 

30 

8.369482 

8.369601 

30 

30 

8.420324 

8.420475 

30 

40 

8.370380 

8.370500 

20 

40 

II23 

1274 

20 

50 

1277 

1397 

10 

50 

1921 

2072 

10 

21 

2171 

2291 

39 

31 

2717 

2869 

29 

10 

3063 

3184 

50 

10 

35" 

3664 

50 

20 

3954 

4076 

40 

20 

4304 

4458 

40 

30 

4843 

4965 

30 

30 

5096 

5250 

30 

40 

573° 

5853 

20 

40 

5886 

6040 

20 

50 

6615 

6738 

10 

50 

6675 

6830 

10 

22 

7499 

7622 

38 

32 

7462 

7618 

28 

10 

8380 

8504 

50 

10 

8248 

8404 

50 

20 

8.379260 

8.379385 

40 

20 

9032 

9189 

40 

30 

8.380138 

8.380263 

30 

30 

8.429815 

8-4*9973 

30 

40 

1015 

1140 

20 

40 

8.430597 

8-43°755 

20 

50 

1889 

2015 

10 

50 

1377 

i536 

10 

23 

2762 

2889 

37 

33 

2156 

23J5 

27 

10 

3633 

3760 

50 

10 

2933 

3°93 

50 

20 

4502 

4630 

40 

20 

37°9 

3870 

40 

30 

5370 

5498 

30 

30 

4484 

4645 

30 

40 

6236 

6364 

20 

40 

5257 

5419 

20 

50 

7100 

7229 

10 

50 

6029 

6191 

10 

24 

7962 

8092 

36 

~3l 

6800 

6962 

26 

10 

8823 

8953 

50 

10 

7569 

7732 

50 

20 

8.389682 

8.389812 

40 

20 

8337 

8500 

40 

30 

40 

8.390539 
1395 

8.390670 
1526 

30 
20 

30 
40 

9103 
8.439868 

8.439267 
8.440033 

30 
20 

50 

2249 

2381 

10 

50 

8.440632 

0797 

10 

25 

3101 

3*34 

35 

35 

1394 

1560 

25 

10 

395i 

4085 

50 

10 

2155 

2322 

50 

20 

4800 

4934 

40 

20 

2915 

3082 

40 

30 

5647 

5782 

30 

30 

3674 

3841 

30 

40 

6493 

6628 

20 

40 

443  1 

4599 

20 

50 

7337 

7472 

10 

50 

5186 

5355 

10 

~26 

8179 

8315 

34 

36 

594i 

6110 

24 

10 

9020 

9156 

50 

10 

6694 

6864 

50 

20 

8.399859 

8.399996 

40 

20 

7446 

7616 

40 

30 

8.400696 

8.400834 

30 

30 

8196 

8367 

30 

40 

i532 

1670 

20 

40 

8946 

9117 

20 

50 

2366 

2505 

10 

50 

8.449694 

8.449866 

10 

27 

3!99 

3338 

33 

37 

8.450440 

8.450613 

23 

10 

4030 

4170 

50 

10 

1186 

J359 

50 

20 

4859 

5000 

40 

20 

1930 

2104 

40 

30 

5687 

5828 

30 

30 

2672 

2847 

30 

40 

6513 

6655 

20 

40 

34*4 

3589 

20 

50 

7338 

7480 

10 

50 

4'54 

433° 

10 

28 

8161 

8304 

32 

38 

4893 

5070 

22 

10 

8983 

9126 

50 

10 

563X 

5808 

50 

20 

8.409803 

8.409946 

40 

20 

6368 

6545 

40 

30 

8.410621 

8.410765 

30 

30 

7103 

7281 

30 

40 

H38 

1583 

20 

40 

7837 

8016 

20 

50 

2254 

2399 

10 

50 

8570 

8749 

10 

29 

3068 

3^13 

31 

39 

8.459301 

8.459481 

21 

10 

3880 

4026 

50 

10 

8.460032 

8.460212 

50 

20 

4691 

4837 

40 

20 

0761 

0942 

40 

30 

55°° 

5647 

30 

30 

1489 

1670 

30 

40 

6308 

6456 

20 

40 

2215 

2398 

20 

50 

7114 

7262 

10 

50 

2941 

3I24 

10 

30 

8.417919 

8.418068 

30 

40 

8.463665 

8.463849 

20 

Cosine. 

Cotang. 

Sec. 

M. 

Cosine. 

Cotang. 

Sec. 

M. 

91°                                       88° 

40 


1°        SINES  AND  TANGENTS.       178° 

M. 

Sec. 

Sine. 

Ts&g. 

M. 

Sec. 

Sine. 

Tang. 

4O 

8.463665 

8.463849 

20 

50 

8.505045 

8.505267 

10 

10 

20 

4388 
5110 

4572 

50 
40 

10 

20 

5702 
6358 

I9o5 
6582 

50 
40 

30 

5830 

6016 

30 

30 

7014 

7238 

30 

40 

655° 

6736 

20 

40 

7668 

7893 

20 

50 

7268 

7455 

10 

50 

8321 

8547 

10 

41 

7985 

8172 

19 

51 

8974 

9200 

9 

10 

8701 

8889 

50 

10 

8.509625 

8.509852 

50 

20 

8.469416 

8.469604 

40 

20 

8.510275 

8.510503 

40 

30 

8.470129 

8.470318 

30 

30 

0925 

II53 

30 

40 

0841 

1031 

20 

40 

1573 

1802 

20 

50 

1553 

J743 

10 

50 

2221 

2451 

10 

42 

2263 

2454 

18 

52 

2867 

3098 

8 

10 

2971 

3l63 

50 

10 

3513 

3744 

50 

20 

3679 

3871 

40 

20 

4J57 

4389 

40 

30 

4386 

4579 

30 

30 

4801 

5°34 

30 

40 
50 

5795 

5^5 
599° 

20 
10 

40 
50 

m 

5677 
6319 

20 
10 

43 

6498 

6693 

17 

53 

6726 

6961 

7 

10 

7200 

7396 

50 

10 

7366 

7602 

50 

20 

7901 

8097 

40 

20 

8005 

8241 

40 

30 

8601 

8798 

30 

30 

8643 

8880 

30 

40 

9299 

8-479497 

20 

40 

9280 

8.519517 

20 

50 

8.479997 

8.480195 

10 

50 

8.519916 

8.520154 

10 

44 

8.480693 

0892 

16 

~54 

8.520551 

0790 

~6~ 

10 

1388 

1588 

50 

10 

1186 

50 

20 

2082 

2283 

40 

20 

1819 

2059 

40 

30 

2775 

2976 

30 

30 

2451 

2692 

30 

40 

3467 

3669 

20 

40 

3083 

33*4 

20 

50 

4158 

4360 

10 

50 

3713 

3956 

10 

45 

4848 

5050 

15 

55 

4343 

4586 

5 

10 

5536 

574° 

50 

10 

4972 

50 

20 

6224 

6428 

40 

20 

5599 

5844 

40 

30 

6910 

7115 

30 

30 

6226 

6472 

30 

40 

7<Q6 

,   7801 

20 

40 

6852 

7098 

20 

50 

T  8280 

8486 

10 

50 

7477 

7724 

10 

~46 

8963 

9170 

14 

~56 

8102 

8349 

4 

10 

8.489645 

8.489852 

50 

10 

8725 

8973 

50 

20 

8.490326 

8-49°534 

40 

20 

9347 

8.529596 

40 

30 

1006 

1215 

30 

30 

8.529969 

8.530218 

30 

40 

1685 

1894 

20 

40 

8.530589 

0840 

20 

50 

2363 

2573 

10 

50 

1209 

1460 

10 

47 

3040 

3250 

13 

57 

1828 

2080 

3 

10 

3715 

3927 

50 

10 

2446 

2698 

50 

20 

439° 

4602 

40 

20 

3063 

3316 

40 

30 

5064 

5276 

30 

30 

3679 

3933 

30 

40 

5736 

5949 

20 

40 

4295 

4549 

20 

50 

6408 

6622 

10 

50 

4909 

5164 

10 

48 

7078 

7293 

12 

58 

5523 

5779 

2  1 

10 

7748 

50 

10 

6136 

6392 

50 

20 

8416 

8632 

40 

20 

6747 

7005 

40 

30 

9084 

9300 

30 

30 

7358 

7616 

30 

40 

8.499750 

8.499967 

20 

40 

7969 

8227 

20 

50 

8.500415 

8.500633 

10 

50 

8578 

8837 

10 

49 

1080 

1298 

11 

59 

9186 

8-539447 

1 

10 

1743 

1962 

50 

10 

8-539794 

8.540055 

50 

20 

2405 

2625 

40 

20 

8.540401 

0662 

40 

30 

3067 

3287 

30 

30 

1007 

1269 

30 

40 
50 

3727 
4386 

3948 
4608 

20 
10 

40 
50 

1612 
2216 

1875 
2480 

20 
10 

50 

8.505045 

8.505267 

10 

60 

8.542819 

8.543084 

0 

Cosine. 

Cotang. 

Sec. 

M. 

Cosine. 

Cotang. 

Sec, 

M. 

91°                                         88° 

41 


0°             LOGARITHMIC           179° 

M. 

Sine.    Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

5iff.  1" 

Cotang. 

0 

Inf.  neg. 

IO.OOOOOO 

.00 

Inf.  neg. 

Infinite. 

60 

1 

6.463726 

5017.17 

o 

6.463726 

5017.17 

13.536274 

59 

2 

764756 

2934.85 

0 

764756 

2934.85 

235244 

58 

3 

6.940847 

2082.31 

0 

6.940847 

2082.31 

I3-°59I53 

57 

4 

5 

7.065786 
162696 

1615.17 
1319.68 

0 

0 

.OO 

7.065786 
162696 

1615.17 
1319.69 

12.934214 
837304 

56 
55 

6 

241877 

1115.78 

9-999999 

.01 

241878 

1115.78 

758122 

54 

7 

308824 

966.53 

99 

308825 

966.54 

691175 

53 

8 

366816 

852.54 

99 

366817 

852.54 

633183 

52 

9 

417968 

762.62 

99 

417970 

762.63 

582030 

51 

!  10 

689188 

98 

463727 

689.88 

536273 

50 

if 

7.505118 

629.81 

9.999998 

7.505120 

629.81 

12.494880 

49 

12 

542906 

579-36 

97 

542909 

579-37 

457091 

48 

13 

577668 

536-41 

97 

577672 

536.42 

422328 

47 

14 

609853 

499.38 

609857 

499-39 

390143 

46 

15 

639816 

467.14 

96 

639820 

467.15 

360180 

45 

16 

667845 

438.81 

95 

667849 

438.82 

332151 

44 

17 

694173 

4I3-72 

95 

694179 

305821 

43 

18 

718997 

94 

719003 

39I-36 

280997 

42 

19 
20 

742477 
764754 

371.27 
353-15 

93 
93 

742484 
764761 

371-28 
353-16 

257516 
235239 

41 

40 

21 

22 

7-785943 
806146 

336.72 

32L75 

9.999992 
91 

7.785951 
806155 

336.73 
321.76 

12.214049 
193845 

39 
38 

23 

825451 

308.05 

90 

.01 

825460 

308.07 

174540 

37 

24 

843934 

295.47 

89 

.02 

843944 

295.49 

156056 

36 

25 

861662 

283.88 

88 

861674 

283.90 

138326 

35 

26 

878695 

273.17 

88 

878708 

273.18 

121292 

34 

27 

_8_a5o8j. 

.  263.23 

_  .  & 

895099 

263.25 

1049.01 

-J3 

28 

"910779 

253.99 

86 

-.."""--  - 

910894 

254.01 

089106 

32 

29 

926119 

245.38 

85 

926134 

245.40 

073866 

31 

30 

940842 

237-33 

83 

940858 

237.35 

059142 

30 

31 

7.955082 

229.80 

9.999982 

7.955100 

229.82 

12.044900 

29 

32 

968870 

222.73 

81 

968889 

222.75 

031111 

28 

33 

982233 

216.08 

80 

982253 

216.10 

017747 

27 

34 

7.995198 

209.81 

79 

7.995219 

209.83 

12.004781 

26 

35 

8.007787 

203.90 

77 

8.007809 

203.92 

11.992191 

25 

36 

020021 

198.31 

76 

020045 

*9*-33 

979955 

24 

37 

031919 

193.02 

75 

031945 

193.05 

968055 

23 

38 

043501 

188.01 

73 

043527 

188.03 

956473 

22 

39 
40 

065776 

183.25 
178.72 

72 

054809 
065806 

183.27 
178.75 

945191 
934194 

21 
20 

Iff 

8.076500 

174.41 

9.999969 

8.076531 

174-44 

11.923469 

19 

42 

086965 

170.31 

68 

086997 

913003 

18 

43 

097183 

166.39 

66 

.02 

097217 

166.42 

902783 

17 

44 

107167 

162.65 

64 

.03 

107202 

162.68 

892798 

16 

45 

116926 

159.08 

63 

116963 

159.11 

883037 

15 

46 

126471 

155.66 

61 

126510 

155.68 

873490 

14 

47 

I358IO 

152.38 

59 

135851 

152.41 

864149 

13 

48 
49 

144953 
153907 

149.24 
146.22 

i 

144996 
153952 

149.27 
146.25 

84604? 

12 
11 

50 

l6268l 

H3-33 

54 

162727 

143.36 

837273 

10 

51 

8.171280 

140.54 

9-999952 

8.171328 

140.57 

11.828672 

9 

52 
53 

I797I3 
187985 

137.86 

48 

179763 
188036 

137.90 
135.32 

820237 
811964 

8 

7 

54 

196102 

132.80 

46 

196156 

132.84 

803844 

6 

55 

204070 

130.41 

44 

.01 

204126 

130.44 

795874 

5 

56 

211895 

128.10 

42 

.O^ 

211953 

128.14 

788047 

4 

57 

219581 

125.87 

40 

219641 

125.91 

780359 

8 

58 

227134 

123.72 

3^ 

227195 

123.76 

772805 

2 

59 

o  234557 

121.  6^: 

36 

.04 

234621 

121.68 

765379 

1 

60 

8.241855 

9-999934 

8.241921 

11.758079 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.]' 

Cotang. 

Diff.  1" 

Tang. 

M. 

90°                                      89° 

42 


1°        SINES  AND  TANGENTS.      178° 

M. 

Sine. 

Diff.  1"  \ 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

~0 

8.241855 

119.63 

9-999934 

.04 

8.241921 

119.67 

11.758079 

60 

1 

249033 

117.68 

93* 

249102 

117.72 

750898 

59 

2 

256094 

115.80 

9*9 

256165 

115.84 

743835 

58 

3 

263042 

113.98 

9*7 

263115 

114.02 

736885 

57 

4 

269881 

112.21 

9*5 

269956 

112.25 

730044 

56 

5 

276614 

110.50 

922 

276691 

110.54 

723309 

55 

6 

283243 

108.83 

920 

*833*3 

108.87 

716677 

54 

7 

289773 

107.22 

917 

289856 

107.26 

710144 

53 

8 

296207 

105.65 

9*5 

296292 

105.70 

703708 

52 

9 

302546 

104.13 

912 

302634 

104.18 

697366 

51 

10 

308794 

102.66 

910 

308884 

102.70 

691116 

50 

11 

8-3H954 

101.22 

9.999907 

8.315046 

101.26 

11.684954 

49 

12 

321027 

99.82 

9°5 

321122 

99.87 

678878 

48 

13 

327016 

98.47 

902 

.04 

327114 

98.51 

672886 

47 

14 

33*9*4 

97.14 

899 

•°5 

3330*5 

97.19 

666975 

46 

15 

338753 

95.86 

897 

338856 

95.90 

661144 

45 

16 

3445°4 

94.60 

894 

344610 

94.65 

655390 

44 

17 

350180 

93-38 

891 

350289 

93-43 

649711 

43 

18 
19 

355783 
361315 

92.19 
91.03 

888 
885 

355895 
361430 

92.24 
91.08 

644105 
638570 

42 
41 

20 

366777 

89.90 

882 

366895 

89-95 

633105 

40 

21 

8.372171 

88.80 

9.999879 

8.372292 

88.85 

11.627708 

~39~ 

22 

377499 

87.72 

876 

377622 

87.77 

622378 

38 

23 

382762 

86.67 

873 

382889 

86.72 

617111 

37 

24 

387962 

85.64 

870 

388092 

85.70 

611908 

36 

25 

393101 

84.64 

867 

393*34 

84.69 

606766 

35 

26 

398179 

83.66 

864 

3983J5 

83.71 

601685 

34 

27 

403199 

82.71 

861 

403338 

82.76 

596662 

33 

28 

408161 

81.77 

858 

408304 

81.82 

591696 

32 

29 

413068 

80.86 

854 

.05 

4132131   80.91 

586787 

31 

30 

417919 

79.96 

851 

.06 

418068]   80.02 

581932 

30 

~31 

8.422717 

79.09 

9.999848 

8.422869 

79.14 

11.577131 

29 

32 

427462 

78.23 

844 

427618 

78.29 

572382 

28 

33 

432156 

77.40 

841 

43*3J5 

77-45 

567685 

27 

34 

436800 

76.57 

838 

436962 

76.63 

563038 

26 

35 

44J394 

75-77 

834 

441560 

75.83 

558440 

25 

36 

445941 

74-99 

831 

446110 

75-°5 

553890 

24 

37 

450440 

74.22 

827 

450613 

74.28 

549387 

23 

38 

454893 

73.46 

823 

455070 

73-5* 

54493° 

22 

39 

459301 

7*-73 

820 

459481 

.   7*-79 

540519 

21 

40 

463665 

72.00 

816 

463849 

72.06 

536l5r 

20 

~£\. 

8.467985 

71.29 

9.999813 

8.468173 

7i-35 

11.531827 

~T9~ 

42 
43 
44 

472263 
476498 
480693 

70.60 
69.91 
69.24 

809 
805 
801 

.06 

47*454 
476693 
480892 

70.66 
69.98 
69.31 

527546 
523307 
519108 

18 
17 
16 

45 

4.8AS4A 

*f  NfctpMf*^ 

•   18J9 

797 

.07 

485050 

68.65 

5J495° 

15 

46 

488963 

67.94 

793 

489170 

68.01 

510830 

14 

47 
48 

493040 
497078 

67.31 
66.69 

790 
786 

493250 
497*93 

67.38 
66.76 

506750 
502707 

13 
12 

49 

501080 

66.08 

782 

501298 

66.15 

498702 

11 

50 

505045 

65.48 

778 

505267 

65-55 

494733 

10 

61 

8.508974 

64.89 

9-999774 

8.509200 

64.96 

11.490800 

9 

52 
53 

512867 
516726 

64.32 
63-75 

769 
765 

513098 
516961 

64.39 
63.82 

486902 
483039 

8 

7 

54 

5*055* 

63.19 

761 

520790 

63.26 

479210 

6 

55 

5*4343 

62.64 

757 

524586 

62.72 

4754H 

5 

56 
57 

528102 
531828 

62.11 
61.58 

753 
748 

528349 
532080 

62.18 
61.65 

471651 
467920 

4 
3 

58 

5355*3 

61.06 

744 

535779 

61.13 

464221 

2 

59 

539186 

60.55 

740 

.07 

539447 

60.62 

460553 

1 

60 

8.542819 

9-999735 

8.543084 

11.456916 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang. 

Diff.  I" 

Tang. 

M. 

|  91°                                     88° 

2°             LOGA 

177° 

M. 

Sine. 

Diff.  1" 

Cosine. 

iff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 
1 
2 

8.542819 
46422 
49995 

60.04 

59-55 
59.06 

9-999735 
731 
726 

.07 
.07 
.07 

8.543084 
46691 
50268 

60.12 
59.62 
59-x4 

11.456916 
53309 
4973* 

60 
59 
58 

3 

53539 

58.58 

722 

.08 

53817 

58.66 

46183 

57 

4 

57°54 

58.11 

717 

57336 

58.19 

42664 

56 

5 

60540 

57-65 

713 

60828 

57-73 

39172 

55 

6 

7 
8 

63999 
67431 
70836 

57-19 
56.74 
56.30 

708 
704 
699 

64291 
67727 
71137 

57.27 
56.82 
56.38 

35709 

32273 

28863 

54 
53 

52 

9 

74214 

55-87 

694 

74520 

55-95 

25480 

51 

10 

77566 

55-44 

689 

77877 

55-5* 

22123 

50 

11 
12 

8.580892 
84193 

55.02 
54.60 

9.999685 
680 

8.581208 
84514 

55-10 
54.68 

11.418792 
15486 

49 

48 

13 

87469 

54-19 

675 

87795 

54-*7 

12205 

4JT 

14 

90721 

53-79 

670 

91051 

53-87 

08949 

46 

15 

93948 

53-39 

665 

94283 

53-47 

05717 

45 

16 

8.597152 

53.00 

660 

8.597492 

53.08 

11.402508 

44 

17 

1.600332 

52.61 

655 

8.600677 

52.70 

11.399323 

43 

18 

19 

06623 

52.23 
51.86 

650 
645 

.08 
.09 

03839 
06978 

52.32 
51.94 

96161 
93022 

42 
41 

20 

09734 

51.49 

640 

10094 

51.58 

89906 

40 

21 
22 

8.612823 
15891 

51.12 

50.76 

9.999635 
629 

8.613189 
16262 

51.21 

50.85 

11.386811 

83738 

39 
38 

23 

18937 

50.41 

624 

i93'3 

50.50 

80687 

37 

24 
25 

21962 
24965 

50.06 
49.72 

619 
614 

22343 
25352 

50-'5 
49.81 

77657 
74648 

36 
35 

26 

27948 

49-38 

608 

28340 

49-47 

71660 

34 

27 

30911 

49.04 

603 

31308 

49.13 

68692 

33 

28 

33854 

48.71 

597 

34256 

48.80 

65744 

32 

29 

36776 

48.39 

59* 

37184 

48.48 

62816 

31 

30 

39680 

48.06 

586 

40093 

48.16 

59907 

30 

31 

8.642563 

47-75 

9.999581 

8.642983 

47-84 

11.357017 

29 

32 

45428 

47-43 

575 

45853 

47-53 

54147 

28 

33 

48274 

47.12 

570 

48704 

47.22 

51296 

27 

34 

51102 

46.82 

564 

".09 

5J537 

46.91 

48463 

26 

35 

53911 

46.52 

558 

.10 

5435* 

46.61 

45648 

25 

36 

56702 

46.22 

553 

57H9 

46.31 

42851 

24 

37 

59475 

45.92 

547 

59928 

46.02 

40072 

23 

38 

62230 

45-63 

54i 

62689 

45-73 

373" 

22 

39 

64968 

45-35 

535 

65433 

45-44 

34567 

21 

40 

67689 

45.06 

5*9 

68160 

45.16 

31840 

20 

~41 

8.670393 

44-79 

9.999524 

8.670870 

44.88 

11.329130 

19 

42 
43 

73080 
75751 

44.51 
44.24 

5l8 
51* 

73563 
76239 

44.61 

44-34 

26437 
23761 

18 
17 

44 

78405 

43-97 

506 

78900 

44.07 

21  100 

16 

45 

8104-: 

43.70 

500 

81544 

43.80 

18456 

15 

46 

83665 

43-4^ 

493 

84172 

43-54 

15828 

14 

47 

86272 

43.18 

487 

86784 

43.28 

13216 

13 

48 

88863 

42.92 

48 

89381 

43-°3 

10619 

12 

49 

9143$ 

42.67 

475 

91963 

42.77 

08037 

11 

50 

93998 

42.42 

469 

.10 

94529 

42.52 

0547I 

10 

61 

96543 

42.17 

9.99946 

.11 

97081 

42.28 

02919 

~~9~ 

52 

8.699073 

41.92 

456 

8.699617 

42.03 

11.300383 

8 

53 

8.701589 

41.6 

450 

8.702139 

41.79 

11.297861 

7 

54 

04090 

41.44 

44 

04646 

41-55 

95354 

6 

55 

06577 

41.2 

43 

07140 

4.1.32 

92860 

5 

56 

09049 

40.9 

43 

09618 

41.08 

90382 

4 

57 

1150 

40.74 

424 

12083 

40.85 

87917 

3 

58 

1195* 

40.5 

41 

H535 

40.62 

85465 

2 

59 

1638 

40.2 

4i 

.1 

16972 

40.40 

83028 

1 

60 

8.71880 

9.999404 

8.719396 

11.280604 

6 

Cosine. 

Diff.  1" 

Sine. 

Diff.l 

Cotang. 

Diff.  V 

Tang. 

M. 

92°                                      87° 

3°        SINES  AND  TANGENTS.       176° 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

8.718800 

40.06 

9.999404 

.11 

8.719396 

40.17 

11.280604 

60 

1 

21204 

39.84 

9398 

21806 

39-95 

78194 

59 

2 

23595 

39.62 

9391 

24203 

39-74 

75797 

58 

3 

25972 

39-4* 

9384 

26588 

39-52 

734*2 

57 

4 

28337 

39-  1  9 

9378 

28959 

39-3* 

71041 

56 

5 

30688 

38.98 

.11 

3*3*7 

39-°9 

68683 

55 

6 

33027 

38.77 

9364 

.12 

33663 

38.89 

66337 

54 

7 

38.57 

9357 

35996 

38.68 

64004 

53 

8 
9 

37667 
39969 

38.36 
38.16 

9350 
9343 

40626 

38.48 
38.27 

61683 
59374 

52 
51 

10 

42259 

37.96 

9336 

42922 

38-07 

57078 

50 

11 

8.744536 

37.76 

9.999329 

8.745207 

37.87 

11.254793 

49 

12 

46802 

37.56 

9322 

47479 

37.68 

52521 

48 

13 

49055 

37-37 

93*5 

49740 

37-49 

50260 

47 

14 

51297 

37.17 

9308 

5*989 

37.29 

48011 

46 

15 

53528 

^6.98 

9301 

54227 

37.10 

45773 

45 

16 

55747 

36.79 

9294 

56453 

36.92 

43547 

44 

17 

18 

57955 
60151 

36.61 
36.42 

9286 
9279 

58668 
60872 

36-73 
36.55 

4*332 
39128 

43 
42 

19 

62337 

36.24 

9272 

63065 

36.36 

36935 

41 

20 

64511 

36.06 

9265 

65246 

36.18 

34754 

40 

~21 

8.766675 

35-88 

9.999257 

.12 

8.767417 

36.00 

11.232583 

39 

22 

68828 

35.70 

9250 

•13 

69578 

35-83 

30422 

38 

23 

70970 

35-53 

9242 

71727 

35-65 

28273 

37 

24 

73101 

35-35 

9235 

73866 

35-48 

26134 

36 

25 

75223 

9227 

75995 

35-3* 

24005 

35 

26 

77333 

35-oi 

9220 

78114 

35'H 

21886 

34 

27 

79434 

34.84 

9212 

80222 

34-97 

19778 

33 

28 
29 
30 

81524 
83605 
85675 

34-67 
34-51 
34-34 

9205 
9197 
9189 

82320 
84408 
86486 

34.80 
34.64 
34-47 

17680 
15592 
*35*4 

32  1 
31 
30 

~3T 

8.787736 

34.18 

9.999181 

8.788554 

34-3* 

11.211446 

29 

32 
33 

89787 
91828 

34.02 
33-86 

9166 

90613 
92662 

34-*5 
33-99 

09387 
07338 

28 
27 

34 

93859 

33-70 

9158 

94701 

33-8^ 

05299 

26 

35 

95881 

33-54 

9150 

96731 

33-68 

03269 

25 

36 

97894 

33-39 

9142 

8.798752 

33-52 

11.201248 

24 

37 

8.799897 

33-23 

9134 

8.800763 

33-37 

11.199237 

23 

38 

8.801892 

33-o8 

9126 

02765 

33-22 

97235 

22 

39 

03876 

32.93 

9118 

04758 

33-07 

95242 

21 

40 

05852 

32.78 

9110 

06742 

32.92 

93258 

20 

41 

8.807819 

32-63 

9.999102 

.13 

8.808717 

32.77 

11.191283 

19 

42 

09777 

32-49 

9094 

.14 

10683 

32.62 

893*7 

18 

43 

11726 

32.34 

9086 

12641 

32.48 

87359 

17 

44 

13667 

32.19 

9077 

14589 

32.33 

854** 

16 

45 

15599 

32-05 

9069 

16529 

32.19 

83471 

15 

46 

17522 

31.91 

9061 

18461 

32.05 

8*539 

14 

47 

19436 

3*-77 

9°53 

20384 

79616 

13 

48 

2*343 

31.63 

9044 

22298 

3*-77 

77702 

12 

49 

23240 

31.49 

9036 

24205 

31.63 

75795 

11 

50 

25130 

3*-35 

9027 

26103 

73897 

10 

51 

8.827011 

31.22 

9.999019 

8.827992 

31.36 

11.172008 

9 

52 

28884 

31.08 

9010 

29874 

31.23 

70126 

8 

53 

30749 

30-95 

9002 

3*748 

31.09 

68252 

7 

54 

32607 

30.82 

8993 

336*3 

30.96 

66387 

6 

55 

34456 

30.69 

8984 

3547* 

30-83 

64529 

5 

56 

36297 

30-56 

8976 

.14 

37321 

30.70 

62679 

4 

57 

38130 

30-43 

8967 

•*5 

39163 

30-57 

60837 

3 

58 

39956 

30.30 

8958 

.15 

40998 

30-45 

59002 

2 

59 

4*774 

30.17 

8950 

•*5 

42825 

30.32 

57*75 

1 

60 

8-843585 

9.998941 

8.844644 

11.155356 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

93°                                       86° 

4°             LOCiARITHlVIIC           175° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

0 

8.843585 

30.05 

9.998941 

•*5 

8.844644 

30.19 

11.155356 

60  I 

1 

45387 

29.92 

932 

46455 

30.07 

53545 

59 

2 

47183 

29.80 

923 

48260 

29.95 

51740 

58 

3 

48971 

29.67 

914 

50057 

29.82 

49943 

57 

4 

50751 

29.55 

905 

51846 

29.70 

48154 

56 

5 

52525 

29-43 

896 

53628 

29.58 

46372 

55 

6 

54291 

29.31 

887 

55403 

29.46 

44597 

54 

7 

56049 

29.19 

878 

57*7* 

29.35 

42829 

53 

8 

57801 

29.08 

869 

58932 

29-23 

41068 

52 

9 

59546 

28.96 

860 

60686 

29.11 

393*4 

51 

10 

61283 

28.84 

85I 

62433 

29.00 

37567 

50 

11 

12 

8.863014 
64738 

28.73 
28.61 

9.998841 
832 

•15 

8.864173 
65906 

28.88 
28.77 

11.135827 
34094 

49 

48 

13 

66455 

28.50 

823 

.16 

67632 

28.66 

32368 

47 

14 

68165 

28.39 

813 

69351 

28.54 

30649 

46 

15 

69868 

28.28 

804 

71064 

28.43 

28936 

45 

16 

71565 

28.17 

795 

72770 

28.32 

27230 

44 

17 

73255 

28.06 

785 

74469 

28.21 

25531 

43 

18 

74938 

27-95 

776 

76162 

28.11 

23838 

42 

19 

76615 

27.84 

766 

77849 

28.00 

22151 

41 

20 

78285 

27-73 

757 

79529 

27.89 

20471 

40 

21 

8.879949 

27.63 

9.998747 

8.881202 

27.79 

11.118798 

39 

22 

81607 

27.52 

738 

82869 

27.68 

17131 

38 

23 

83258 

27.42 

728 

84530 

27.58 

15470 

37 

24 

84903 

27.31 

718 

86185 

27.47 

13815 

36 

25 

86542 

27.21 

708 

87833 

27-37 

12167 

35 

26 

88174 

27.11 

699 

89476 

27.27 

10524 

34 

27 

89801 

27.00 

689 

91112 

27.17 

08888 

33 

28 
29 

91421 
93035 

26.90 
26.80 

679 
669 

.16 

•17 

92742 
94366 

27.07 
26.97 

07258 
05634 

32 
31 

30 

94643 

26.70 

659 

95984 

26.87 

04016 

30 

32 
33 

96245 
97842 
8.899432 

26.60 
26.51 
26.41 

9.998649 

639 
629 

97596 
8.899203 
8.900803 

26.77 
26.67 
26.58 

02404 
11.100797 
11.099197 

29 

28 
27 

34 

8.901017 

26.31 

619 

02398 

26.48 

97602 

26 

35 

02596 

26.22 

609 

03987 

26.38 

96013 

25 

36 

04169 

26.12 

599 

05570 

26.29 

94430 

24 

37 

05736 

26.03 

589 

07147 

26.20 

92853 

23 

38 

07297 

25.93 

578 

08719 

26.10 

91281 

22 

39 

08853 

25.84 

568 

10285 

26.01 

89715 

21 

40 

10404 

25-75 

558 

11846 

25.92 

88154 

20 

41 

8.911949 

25.66 

9.998548 

8.913401 

25.83 

11.086599 

19 

42 

13488 

25-56 

537 

14951 

25.74 

85049 

18 

43 

15022 

25.47 

527 

•17 

16495 

25.65 

83505 

17 

44 

16550 

25.38 

516 

.18 

18034 

25.56 

81966 

16 

45 

18073 

25.29 

506 

19568 

2547 

80432 

15 

46 

I959I 

25.20 

495 

21096 

25.38 

78904 

14 

47 

21103 

25.12 

485 

22619 

25.30 

13 

48 

22610 

25.03 

474 

24136 

25.21 

75864 

12 

49 

24112 

24.94 

464 

25649 

25.12 

7435* 

11 

50 

25609 

24.86 

453 

27156 

25.03 

72844 

10 

51 

8.927100 

24.77 

9.998442 

8.928658 

24.95 

11.071342 

9 

52 

28587 

24.69 

43* 

3OI55 

24.86 

69845 

8 

53 

30068 

24.60 

421 

3*647 

24.78 

7 

54 
55 

3*544 
33OI5 

24-52 
24.43 

410 
399 

33*34 
34616 

24.70 
24.61 

66866 
65384 

6 
5 

56 

34481 

24.35 

388 

36093 

24.53 

63907 

4 

57 

35942 

24.27 

377 

37565 

24.45 

62435 

3 

58 

37398 

24.19 

366 

39032 

24.37 

60968 

2 

59 
60 

38850 
8.940296 

24.11 

355 
9.998344 

.18 

40494 
8.941952 

24.29 

59506 
11.058048 

1 
0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  V 

Tang. 

M. 

94°                                        85° 

46 


5°        SINES  AND  TAWCHENTS.      174° 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  I" 

Cotang. 

0 

8.940296 

24.03 

9.998344 

.19 

8.941952 

24.21 

11.058048 

60 

1 

41738 

23-94 

333 

43404 

24.13 

56596 

59 

2 
3 

44606 

23.87 
23.79 

322 
3" 

44852 
46295 

24.05 
23-97 

55H8 

53705 

58 
57 

4 

46034 

23.71 

3°o 

47734 

27.90 

52266 

56 

5 

47456 

23.63 

289 

49168 

23.82 

50832 

55 

6 

48874 

23-55 

277 

50597 

23-74 

49403 

54 

7 

50287 

23.48 

266 

52021 

23.66 

47979 

53 

8 

51696 

23.40 

255 

53441 

23.59 

46559 

52 

9 

53100 

23-32 

243 

54856 

23-5I 

45H4 

51 

10 

54499 

23.25 

232 

56267 

23-44 

43733 

50 

11 

8.955894 

23.17 

9.998220 

8.957674 

23-37 

11.042326 

49 

12 

57284 

23.10 

209 

59075 

23-29 

40925 

48 

13 

58670 

23.02 

197 

60473 

23.22 

39527 

47 

14 

60052 

22.95 

186 

61866 

23.14 

38134 

46 

15 

61429 

22.88 

174 

63255 

23.07 

36745 

45 

16 

62801 

22.80 

163 

64639 

23.00 

35361 

44 

17 

64170 

22.73 

.19 

66019 

22.93 

33981 

43 

18 

65534 

22.66 

139 

.20 

67394 

22.86 

32606 

42 

19 

66893 

22.59 

128 

68766 

22.79 

3I234 

41 

20 

68249 

22.52 

116 

70133 

22.71 

29867 

40 

21 

8.969600 

22.45 

9.998104 

8.971496 

22.65 

11.028504 

39 

22- 

70947 

22.38 

092 

72855 

22.57 

27145 

38 

23 

72289 

22.31 

080 

74209 

22.51 

25791 

37 

24 

73628 

22.24 

068 

7556o 

22.44 

24440 

36 

25 

74962 

22.17 

056 

76906 

22.37 

23094 

35 

26 
27 

76293 
77619 

22.10 
22.03 

044 
032 

78248 
79586 

22.30 

22.23 

21752 
20414 

34 
33 

28 

78941 

21.97 

020 

80921 

22.17 

19079 

32 

29 

80259 

21.90 

9.998008 

82251 

22.10 

17749 

31 

30 

21.83 

9.997996 

83577 

22.04 

16423 

30 

31 

8.982883 

21.77 

984 

8.984899 

21-97 

11.015101 

29 

32 

84189 

21.70 

972 

86217 

21.91 

13783 

28 

33 

85491 

21.63 

959 

87532 

21.84 

12468 

27 

34 

86789 

21-57 

947 

.20 

88842 

21.78 

11158 

26 

35 

88083 

21.50 

935 

.21 

90149 

21.71 

09851 

25 

36 

89374 

21.44 

922 

91451 

21.65 

08549 

24 

37 

90660 

21.38 

910 

92750 

21.58 

07250 

23 

38 

91943 

21.31 

897 

94045 

21.52 

05955 

22 

39 
40 

93222 
94497 

21.25 
21.19 

885 
872 

95337 
96024 

21.46 
21.40 

04663 
03376 

21 
20 

Itl 

8.995768 

21.12 

9.997860 

97908 

21.34 

02092 

19 

42 

97036 

21.06 

847 

8.999188 

21.27 

11.000812 

18 

43 

98299 

21.  OO 

835 

9.000465 

21.21 

10-999535 

17 

44 

8.999560 

20.94 

822 

01738 

21.15 

98262 

16 

45 

9.000816 

20.88 

809 

03007 

21.09 

96993 

15 

46 

02069 

20.82 

797 

04272 

21.03 

95728 

14 

47 

03318 

20.76 

784 

05534 

20.97 

94466 

13 

48 

04563 

20.70 

771 

06792 

20.91 

93208 

12 

49 

05805 

20.64 

758 

08047 

20.85 

9'953 

11 

50 

07044 

20.58 

745 

09298 

20.80 

90702 

10 

51 

52 

9.008278 
09510 

20.52 
20.46 

9.997732 
719 

9.010546 
11790 

20.74 

20.68 

10.989454 
'88210 

9 

8 

53 

10737 

20.40 

706 

.21 

13031 

20.62 

86969 

7 

54 

11962 

20.34 

693 

.22 

14268 

20.56 

85732 

6 

55 

13182 

20.29 

680 

15502 

20.51 

84498 

5 

56 

14400 

20.23 

667 

16732 

20.45 

83268 

4 

57 

15613 

20.17 

654 

17959 

20.40 

82041 

3 

58 

16824 

20.12 

641 

• 

19183 

20.34 

80817 

2 

59 

18031 

20.06 

628 

.22 

20403 

20.28 

79597 

1 

60 

9.019235 

9.997614 

9.021620 

10.978380 

0 

Cosine. 

Diff.  1" 

Sine. 

Difif.1" 

Cotang. 

Diff.  1" 

Tang. 

M. 

i  95°                                       84° 

47 


6°            LOGA 

173°  ! 

RITXX1VCIC 

M. 

Sine. 

Diff.  V 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.019235 

2O.OO 

9.997614 

.22 

9.021620 

20.23 

10.978380 

60 

1 

20435 

19-95 

601 

22834 

20.17 

77166 

59 

2 

21632 

19.89 

588 

24044 

20.11 

75956 

58 

3 

22825 

19.84 

574 

*5*5' 

2O.O6 

74749 

57 

4 

24016 

19.78 

561 

*6455 

20.01 

73545 

56 

5 

25203 

19-73 

547 

.22 

27655 

19.95 

7*345 

55 

6 

26386 

19.67 

534 

•*3 

28852 

19.90 

71148 

54 

7 

27567 

19.62 

520 

30046 

19.85 

69954 

53 

8 

28744 

19-57 

5°7 

3I237 

19.79 

68763 

52 

9 

29918 

'9-51 

493 

3*4*5 

19.74 

67575 

51 

10 

31089 

19.46 

480 

33609 

19.69 

66391 

50 

11 
12 

9.032257 
334*i 

19.41 

19.36 

9-997466 
45* 

9.034791 
35969 

19.64 
19.58 

10.965209 
64031 

49 

48 

13 

34582 

19.30 

439 

37144 

19-53 

62856 

47 

14 

35741 

19.25 

4*5 

38316 

19.48 

61684 

46 

15 

36896 

19.20 

411 

39485 

19.43 

60515 

45 

16 

38048 

I9-I5 

397 

40651 

19.38 

59349 

44 

17 

39197 

19.10 

383 

41813 

19-33 

58187 

43 

18 

40342 

19.05 

369 

4*973 

19.28 

57027 

42 

19 

41485 

18.99 

355 

44I3° 

19.23 

55870 

41 

20 

42625 

18.95 

.23 

45284 

19.18 

54716 

40 

21 

9.043762 

18.89 

9-9973*7 

.24 

9.046434 

19.13 

10.953566 

39 

22 

44895 

18.84 

47582 

19.08 

52418 

38 

23 

46026 

18.79 

299 

487*7 

19.03 

5i*73 

37 

24 

47*54 

18.75 

285 

49869 

18.98 

50131 

36 

25 

48279 

18.70 

271 

51008 

18.93 

48992 

35 

26 

49400 

18.65 

257 

52144 

18.89 

47856 

34 

27 

5°5X9 

18.60 

242 

53*77 

18.84 

46723 

33 

28 

5l635 

18.55 

228 

544°7 

I8.79 

45593 

32 

29 

5*749 

18.50 

214 

55535 

18.74 

44465 

31 

30 

53859 

18.45 

199 

56659 

18.70 

43341 

30 

31 

9.054966 

18.41 

9.997185 

9.057781 

18.65 

10.942219 

29 

32 

56071 

18.36 

170 

58900 

18.60 

41100 

28 

33 

5717* 

18.31 

156 

60016 

18.55 

39984 

27 

34 

58271 

18.27 

141 

61130 

18.51 

38870 

26 

35 

59367 

18.22 

127 

62240 

18.46 

37760 

25 

36 

60460 

18.17 

112 

63348 

18.42 

36652 

24 

37 

61551 

18.13 

098 

.24 

64453 

18.37 

35547 

23 

38 

•  62639 

18.08 

083 

.25 

65556 

18.33 

34444 

22 

39 
40 

637*4 
64806 

18.04 
17.99 

068 

°53 

66655 

67752 

18.28 
18.24 

33345 
32248 

21 
20 

41 

9.065885 

17.94 

9.997039 

9.068846 

18.19 

10.931154 

19 

42 

66962 

17.90 

024 

69938 

18.15 

30062 

18 

43 

68036 

17.86 

9.997009 

71027 

18.10 

28973 

17 

44 

69107 

17.81 

9-996994 

72113 

18.06 

27887 

16 

45 

70176 

17.77 

979 

73197 

18.02 

26803 

15 

46 

71242 

17.72 

964 

74*78 

17.97 

25722 

14 

47 

72305 

17.68 

949 

75356 

J7-93 

24644 

13 

48 

73366 

17.63 

934 

76432 

17.89 

23568 

12 

49 

744*4 

17-59 

919 

775°5 

17.8^ 

22495 

11 

50 

75480 

17-55 

904 

78576 

17.80 

21424 

10 

"51 

9.076531 

17-5° 

9.996889 

9.079644 

17.76 

10.920356 

9 

52 

17.46 

874 

80710 

17.72 

19290 

8 

53 

78635 

17.42 

858 

8i773 

17.67 

18227 

7 

54 

79676 

I7-38 

843 

82833 

17.63 

17167 

6 

55 

80719 

17-33 

828 

.25 

83891 

17-59 

16109 

5 

56 

8i759 

17.29 

812 

.26 

84947 

17-55 

J5°5' 

4 

57 

82797 

17.25 

797 

86000 

17.51 

14000 

3 

58 

83832 

17.21 

782 

87050 

17-47 

12950 

2 

59 

84864 

17.17 

766 

.26 

88098 

17-43 

11902 

1 

60 

9.08589^ 

9.996751 

9.08914^ 

10.910856 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l' 

Cotang. 

Diff.  1" 

Tang. 

M. 

96°                                        83° 

48 


7°       SINES  AND  TANGENTS.      172° 

M. 

Sine. 

Diff.  V 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.085894 

17.13 

9.996751 

.26 

9.089144 

I7-38 

10.910856 

60 

1 

86922 

17.09 

735 

90187 

17-35 

09813 

59 

2 

87947 

17.04 

720 

91228 

17.30 

08772 

58 

3 

88970 

17.00 

92266 

17.27 

07734 

57 

4 

89990 

16.96 

688 

93302 

17.22 

06698 

56 

5 

91008 

16.92 

673 

94335 

17.19 

05664 

55 

6 

92024 

16.88 

657 

95367 

17.15 

04633 

54 

7 

93037 

16.84 

64I 

96395 

03605 

53 

8 

94047 

16.80 

625 

97422 

17.07 

02578 

52 

9 

10 

95056 
96062 

16.76 
16.73 

610 

594 

.26 

98446 
99468 

17.03 
16.99 

01554 
10.900532 

51 

50 

11 
12 

9.097065 
98066 

16.68 
16.65 

9.996578 
562 

.27 

9.100487 
01504 

16.95 
16.91 

10.899513 
98496 

49 
48 

13 

9.099065 

16.61 

546 

02519 

16.87 

97481 

47 

14 

9.100062 

16.57 

53° 

03532 

16.84 

96468 

46 

15 

01056 

16.53 

5H 

04542 

16.80 

95458 

45 

16 

02048 

16.49 

498 

0555o 

16.76 

94450 

44 

17 
18 
19 

03037 
04025 
05010 

16.45 
16.42 
16.38 

482 
465 
449 

06556 

07559 
08560 

16.72 
16.69 
16.65 

93444 
92441 
91440 

43 
42 
41 

20 

05992 

16.34 

433 

°9559 

16.61 

90441 

40 

21 

9.106973 

16.30 

9.996417 

9.110556 

16.58 

10.889444 

39 

22 

07951 

16.27 

400 

11551 

16.54 

88449 

38 

23 

08927 

16.23 

384 

12543 

16.50 

87457 

37 

24 

09901 

16.19 

368 

J3533 

16.47 

86467 

36 

25 

10873 

16.16 

351 

14521 

16.43 

85479 

35 

26 

11842 

16.12 

335 

15507 

16.39 

84493 

34 

27 

12809 

16.08 

.27 

16491 

16.36 

33 

28 

13774 

16.05 

302 

.28 

17472 

16.32 

82528 

32 

29 

14737 

16.01 

285 

18452 

16.29 

81548 

31 

30 

15698 

15-97 

269 

19429 

16.25 

80571 

30 

IT 

9.116656 

15.94 

9.996252 

9.120404 

16.22 

10.879596 

29 

32 

17613 

15.90 

235 

21377 

16.18 

78623 

28 

33 

18567 

15.87 

219 

22348 

16.15 

77652 

27 

34 

19519 

15.83 

202 

23317 

16.11 

76683 

26 

35 

20469 

15.80 

185 

24284 

16.08 

75716 

25 

36 

21417 

15.76 

168 

25249 

16.04 

74751 

24 

37 

22362 

15-73 

151 

26211 

16.01 

73789 

23 

38 

23306 

15.69 

134 

27172 

15-97 

72828 

22 

39 

24248 

15.66 

117 

28130 

*5-94 

71870 

21 

40 

25187 

15.62 

100 

.28 

29087 

15.91 

70913 

20 

~4l 

9.126125 

15-59 

9.996083 

.29 

9.130041 

15.87 

10.869959 

19 

42 

27060 

I5-56 

066 

3°994 

15.84 

69006 

18 

43 

27993 

I5-52 

049 

31944 

15.81 

68056 

17 

44 

28925 

15.49 

032 

32893 

15-77 

67107 

16 

45 

29854 

15-45 

9.996015 

33839 

15-74 

6616; 

15 

46 

30781 

15.42 

9.995998 

34784 

15.71 

65216 

14 

47 

31706 

15-39 

980 

35726 

15.67 

64274 

13 

48 

32630 

15-35 

963 

36667 

15.64 

63333 

12 

49 

33551 

I5-32 

946 

37605 

15.61 

62395 

11 

50 

3447° 

15.29 

928 

38542 

15.58 

61458 

10 

~5T 

9-I35387 

15.25 

9.995911 

9.139476 

15-55 

10.860524 

9 

52 

36303 

15.22 

894 

40409 

15.51 

59591 

8 

53 

37216 

15.19 

876 

41340 

15.48 

58660 

7 

54 

38128 

15.16 

859 

42269 

57731 

6 

55 

39037 

15.12 

841 

43196 

15.42 

56804 

5 

56 

39944 

15.09 

823 

44121 

15-39 

55879 

4 

57 

40850 

15.06 

806 

45044 

15-35 

54956 

3 

58 

4J754 

15-03 

788 

45966 

I5-32 

54034 

2 

59 

42655 

15.00 

771 

.29 

46885 

15.29 

53"5 

1 

60 

9-H3555 

9-995753 

9.147803 

10.852197 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

97°                                      82° 

49 


8°             LOGARITHMIC           171°  1 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9-143555 

14.96 

9-995753 

.30 

9.147803 

15.26 

10.852197 

60 

1 

4453 

14-93 

735 

8718 

15.23 

1282 

59 

2 

5349 

14.90 

717 

9.149632 

15.20 

10.850368 

58 

3 

6243 

14.87 

699 

9.150544 

I5-I7 

10.849456 

57 

4 

7136 

14.84 

681 

1454 

8546 

56 

5 

8026 

14.81 

664 

2363 

15.11 

7637 

55 

6 

8915 

14.78 

646 

3269 

15=08 

6731 

54 

7 

9.149802 

14.75 

628 

4174 

I5-05 

5826 

53 

8 

9.150686 

14.72 

610 

5077 

15.02 

49*3 

52 

9 

1569 

14.69 

591 

14.99 

4022 

51 

10 

2451 

14.66 

573 

6877 

14.96 

3123 

50 

11 

9-15333° 

14.63 

9-995555 

9-157775 

14-93 

10.842225 

49 

12 

4208 

14.60 

537 

8671 

14.90 

1329 

48 

13 

5°83 

14-57 

•3° 

9.159565 

14.87 

10.840435 

47 

14 

5957 

14.54 

501 

•31 

9.160457 

14.84 

10.839543 

46 

15 

6830 

14.51 

482 

1347 

14.81 

8653 

45 

16 

7700 

14.48 

464 

2236 

14.78 

7764 

44 

17 

8569 

14.45 

446 

3123 

'4-75 

6877 

43 

18 

9-'59435 

14.42 

427 

4008 

14.73 

599* 

42 

19 

9.160301 

14-39 

409 

4892 

14.70 

*   5108 

41 

20 

1164 

14.36 

39° 

5774 

14.67 

4226 

40 

22 

9.162025 
2885 

14-33 
14.30 

9-99537* 
353 

9.166654 

753* 

14.64 
14.61 

10.833346 

39 
38 

23 

3743 

14.27 

334 

8409 

14.58 

1591 

37 

24 

4600 

14.24 

316 

9.169284 

14-55 

10.830716 

36 

25 

5454 

14.22 

297 

9.170157 

J4-53 

10.829843 

35 

26 

6307 

14.19 

278 

1029 

14.50 

8971 

34 

27 

7159 

14.16 

260 

•31 

1899 

14.47 

8101 

33 

28 

8008 

I4-I3 

241 

•3* 

2767 

14.44 

7*33 

32 

29 

8856 

14.10 

222 

3634 

14.42 

6366 

31 

30 

9.169702 

14.07 

203 

4499 

J4-39 

5501 

30 

31 

9.170547 

14.05 

9-99*5  1  84 

9.175362 

14.36 

10.824638 

29 

32 

1389 

14.02 

6224 

I4-33 

3776 

28 

33 

2230 

13-99 

146 

7084 

H-31 

2916 

27 

34 

3070 

13.96 

127 

7942 

14.28 

2058 

26 

35 

3908 

13-94 

108 

8799 

14.25 

1201 

25 

36 

4744 

13.91 

089 

9.179655 

*4-*3 

10.820345 

24 

37 

5578 

13.88 

070 

9.180508 

14.20 

10.819492 

23 

38 

6411 

13.86 

051 

1360 

14.17 

8640 

22 

39 

7242 

13.83 

032 

2211 

14.15 

7789 

21 

40 

8072 

13.80 

9.995013 

3059 

14.12 

6941 

20 

41 

8900 

13-77 

9-994993 

9.183907 

14.09 

10.816093 

19 

42 

9.179726 

13-74 

974 

475* 

14.07 

5*48 

18 

43 

9.180551 

13.72 

955 

5597 

14.04 

4403 

17 

44 

1374 

13.69 

935 

.32 

6439 

14.02 

3561 

16 

45 

2196 

13.67 

916 

•33 

7280 

13-99 

2720 

15 

46 

3016 

13.64 

896 

8120 

13.96 

1880 

14 

47 

3834 

13.61 

877 

8958 

13.94 

1042 

13 

48 

J3-59 

857 

9.189794 

I3-9I 

I0.8I0206 

12 

49 

5466 

I3-56 

838 

9.190629 

13.89 

10.809371 

11 

50 

6280 

13-53 

818 

1462 

13.86 

8538 

10 

51 

9.187092 

13.51 

9-994798 

9.192294 

13.84 

10.807706 

9 

52 

7903 

13.48 

779 

3'*4 

13.81 

6876 

8 

53 

8712 

13.46 

759 

3953 

13-79 

6047 

7 

54 

9.189519 

13-43 

739 

4780 

13.76 

5220 

6 

55 

9.190325 

I3-4I 

719 

5606 

13-74 

4394 

5 

56 

1130 

I3-38 

700 

6430 

13.71 

357° 

4 

57 

*933 

13.36 

68c 

7*53 

13.69 

2747 

3 

58 

*734 

13-33 

660 

8074 

13.66 

1926 

2 

59 

3534 

13.30 

640 

•33 

8894 

13.64 

1106 

1 

60 

9.194332 

9.994620 

9.199713 

10.800287 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

98°                               81° 

50 


,  9°        SINES  AND  TANGENTS.      170° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

~0 

9.194332 

13.28 

9.994620 

•33 

9.199713 

13.61 

10.800287 

60 

1 

5"9 

13.26 

600 

•33 

9.200529 

13-59 

10.799471 

59 

2 

5925 

13.23 

580 

•33 

1345 

I3-56 

8655 

58 

3 

6719 

13.21 

56o 

•34 

2159 

'3-54 

7841 

57 

4 

75" 

13.18 

540 

2971 

7029 

56 

5 

8302 

13.16 

519 

3782 

13.49 

6218 

55 

6 

9091 

I3-I3 

499 

4592 

13-47 

5408 

54 

7 

9.199879 

479 

5400 

13-45 

4600 

53 

8 

9.200666 

13.08 

459 

6207 

13.42 

3793 

52 

9 

1451 

13.06 

438 

7013 

13.40 

2987 

51 

10 

2234 

13.04 

418 

7817 

13.38 

2183 

50 

11 

9.203017 

13.01 

9-994397 

8619 

13-35 

1381 

49 

12 

3797 

12.99 

377 

9.209420 

10.790580 

48 

13 

4577 

12.96 

357 

9.210220 

I3-31 

10.789780 

47 

14 

5354 

12.94 

336 

1018 

13.28 

8982 

46 

15 

6131 

12.92 

316 

1815 

13.26 

8185 

45 

16 

6906 

12.89 

295 

•34 

2611 

13.24 

7389 

44 

17 

7679 

12.87 

274 

•35 

3405 

13.21 

6595 

43 

18 

8452 

12.85 

254 

4198 

13.19 

5802 

42 

19 

9222 

12.82 

4989 

5011 

41 

20 

9.209992 

12.80 

212 

578o 

13.15 

4220 

40 

21 

9.210760 

12.78 

9.994191 

9.216568 

13.12 

10.783432 

39 

22 

1526 

"•75 

171 

7356 

13.10 

2644 

38 

23 

2291 

12.73 

150 

8142 

13.08 

1858 

37 

24 

3°55 

12.71 

129 

8926 

13.05 

1074 

36 

25 

3818 

12.68 

1  08 

9.219710 

13.03 

10.780290 

35 

26 

4579 

12.66 

087 

9.220492 

13.01 

10.779508 

34 

27 

533s 

12.64 

066 

1272 

12.99 

8728 

33 

28 

6097 

12.  6l 

045 

2052 

12.97 

7948 

32 

29 

6854 

12.59 

024 

2830 

12.94 

7170 

31 

30 

7609 

12.57 

9.994003 

3606 

12.92 

6394 

30 

31 

32 

9.218363 
9116 

12.55 

"•53 

9.993981 
960 

9.224*382 
5156 

12.90 
12.88 

10.775618 
4844 

29 

28 

33 

9.21986$ 

12.50 

939 

5929 

12.86 

4071 

27 

34 

9.220618 

12.48 

918 

•35 

6700 

12.84 

3300 

26 

35 

1367 

12.46 

896 

.36 

7471 

12.81 

2529 

25 

36 

2115 

12.44 

875 

8239 

12.79 

1761 

24 

37 

2861 

12.42 

8*4 

9007 

12.77 

0993 

23 

38 

3606 

12.39 

832 

9.229773 

12.75 

10.770227 

22 

39 

4349 

12.37 

811 

9-230539 

12.73 

10.769461 

21 

40 

5092 

12.35 

789 

1302 

12.71 

8698 

20 

41 

9.225833 

"•33 

9.993768 

9.232065 

12.69 

10.767935 

19 

42 

6573 

12.31 

746 

2826 

12.67 

7174 

18 

43 

7311 

12.28 

725 

3586 

12.65 

6414 

17 

44 

8048 

12.26 

7°3 

4345 

12.62 

5655 

16 

45 

8784 

12.24 

681 

5103 

12.60 

4897 

15 

46 

9.229518 

12.22 

660 

5859 

12.58 

4141 

14 

47 

9.230252 

12.20 

638 

661^ 

12.56 

3386 

13 

48 

0984 

12.18 

616 

•36 

7368 

12.54 

2632 

12 

49 

1714 

12.16 

594 

•37 

8120 

12.52 

1880 

11 

50 

2444 

12.1^; 

57^ 

8872 

12.50 

1128 

10 

~51 

9.233172 

12.12 

9-993550 

9.239622 

12.48 

10.760378 

9 

52 

3899 

12.09 

528 

9.240371 

12.46 

10.759629 

8 

53 

4625 

12.07 

506 

1118 

12.44 

8882 

7 

54 

5349 

12.05 

484 

1865 

12.42 

8135 

6 

55 

607; 

12.0^ 

462 

2610 

12.40 

739° 

5 

56 

6795 

12.01 

440 

3354 

12.38 

6646 

4 

57 

75'5 

11-99 

418 

4097 

12.36 

59°3 

3 

58 

8235 

11-97 

396 

4839 

12.34 

5161 

2 

59 

8953 

11-95 

374 

•37 

5579 

12.32 

4421 

1 

60 

9.239670 

9.993351 

9.246319 

10.753681 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l' 

Cotang. 

Diff.  1" 

Tang. 

M. 

99°                                        80° 

51 


10°            XiOCKA 

169° 

HITHBrllC 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.239670 

11.93 

9-993351 

•37 

9.246319 

12.30 

10.753681 

60  j 

1 

9.240386 

11.91 

329 

7057 

12.28 

2943 

59  I 

2 

IIOI 

11.89 

307 

7794 

12.26 

2206 

58  : 

3 

1814 

11.87 

285 

8530 

12.24 

1470 

57 

4 

2526 

11.85 

262 

9264 

12.22 

0736 

56 

5 

3237 

11.83 

240 

•37 

9.249998 

12.20 

10.750002 

55 

6 

3947 

ii.  81 

217 

.38 

9.250730 

12.18 

10.749270 

54 

7 

4656 

11.79 

195 

1461 

12.17 

8<39 

53 

8 

5363 

11.77 

172 

2191 

12.15 

7809 

52 

9 

6069 

11.75 

149 

2920 

12.13 

7080 

51 

10 

6775 

11.73 

I27 

3648 

12.11 

6352 

50 

11 

9.247478 

11.71 

9.993104 

9.254374 

12.09 

10.745626 

49 

12 

8181 

11.69 

081 

5100 

12.07 

4900 

48 

13 

8883 

11.67 

059 

5824 

12.05 

4176 

47 

14 
15 

9.249583 
9.250282 

11.65 
11.63 

036 
9.993013 

6547 
7269 

I2.O3 
12.01 

3453 
2731 

46 
45 

16 

0980 

n.  61 

9.992990 

7990 

12.00 

2010 

44 

17 

18 

1677 
*373 

11.59 
11.58 

967 
944 

8710 
9.259429 

11.98 
11.96 

1290 
10.740571 

43 
42 

19 

3067 

11.56 

921 

9.260146 

11.94 

10.739854 

41 

20 

3761 

11.54 

898 

0863 

11.92 

9'37 

40 

~21 

9-254453 

11.52 

9.992875 

9.261578 

11.90 

10.738422 

~3iT 

22 
23 

5*44 

5834 

11.50 
11.48 

852 
829 

•38 
•39 

2292 
3005 

11.89 

11.87 

7708 
6995 

38 
37 

24 

6523 

11.46 

806 

3717 

11.85 

6283 

36 

25 

7211 

11.44 

783 

4428 

11.83 

557* 

35 

26 

7898 

11.42 

759 

5138 

II.«1 

4862 

34 

27 

8582 

11.41 

736 

5847 

11-79 

4153 

33 

28 

9268 

"•39 

713 

6555 

II.78 

3445 

32 

29 

9.259951 

11.37 

690 

7261 

11.76 

*739 

31 

30 

9.260633 

"•35 

666 

7967 

11-74 

2033 

30 

«I 

32 

1314 
1994 

"•33 
11.31 

9.992643 
619 

8671 
9.269375 

11.72 

11.70 

1329 
10.730625 

29 
28 

33 

2673 

11.30 

596 

9.270077 

11.69 

10.729923 

27 

34 

3351 

11.28 

572 

0779 

11.67 

9221 

26 

35 

4027 

11.26 

549 

1479 

11.65 

8521 

25 

36 

4703 

11.24 

5*5 

2178 

11.64 

7822 

24 

37 

5377 

11.22 

5°' 

•39 

»  2876 

11.62 

7124 

23 

38 

6051 

11.20 

478 

.40 

3573 

II.  60 

6427 

22 

39 

6723 

11.19 

454 

4269 

11.58 

5731 

21 

40 

7395 

11.17 

430 

4964 

11.57 

5036 

20 

41 

9.268065 

11.15 

9.992406 

9.275658 

11.55 

10.724342 

19 

42 

8734 

11.13 

382 

6351 

11.53 

3649 

18 

43 

9.269402 

II.  12 

359 

7°43 

11.51 

2957 

17 

44 

9.270069 

II.  IO 

335 

7734 

11.50 

2266 

16 

45 

°735 

11.08 

3" 

8424 

11.48 

1576 

15 

46 

1400 

1  1.  06 

287 

9"3 

11.46 

0887 

14 

47 

2064 

11.05 

263 

9.279801 

11-45 

10.720199 

13 

48 

2726 

11.03 

239 

9.280488 

"•43 

10.719512 

12 

49 

3388 

II.OI 

214 

"74 

11.41 

8826 

11 

50 

4049 

10.99 

190 

1858 

11.40 

8142 

10 

51 

9.274708 

10.98 

9.992166 

9.282542 

11.38 

10.717458 

9 

52 

5367 

10.96 

142 

.40 

3225 

11.36 

6775 

8 

53 

6024 

10.94 

117 

.41 

39°7 

"•35 

6093 

7 

54 

6681 

10.92 

093 

4588 

"•33 

5412 

6 

55 

7337 

10.91 

069 

5268 

11.31 

4732 

5 

56 
57 

7991 

8644 

10.89 
10.87 

044 
9.992020 

6624 

11.30 
11.28 

4°53 
3376 

4 
3 

58 

9297 

10.86 

9.991996 

7301 

11.26 

2699 

2 

59 

9.279948 

10.84 

971 

.41 

7977 

11.25 

2023 

1 

60 

9.280599 

9.991947 

9.288652 

10.711348 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang. 

Diff.  V 

Tang. 

M. 

100°                                      79° 

52 


11°       SINES  A1TO  TANCfflNTTS.       168° 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.280599 

10.82 

9.991947 

-41 

9.288652 

11.23 

10.711348 

60 

1 

1248 

I0.8l 

922 

9326 

11.22 

0674 

59 

2 

1897 

10.79 

897 

9.289999 

11.20 

10.710001 

58 

3 

2544 

10.77 

873 

9.290671 

II.  18 

10.709329 

57 

4 

3190 

10.76 

848 

1342 

11.17 

8658 

56 

5 

3836 

10.74 

823 

2013 

11.15 

7987 

55 

6 

4480 

10.72 

799 

.41 

2682 

11.14 

7318 

54 

7 

5I2A 

10.71 

774 

.42 

3350 

II.  12 

6650 

53 

8 

5766 

10.69 

749 

4017 

II.  II 

5983 

52 

9 

6408 

10.67 

724 

4684 

11.09 

53l6 

51 

10 

7048 

10.66 

699 

5349 

11.07 

4651 

50 

11 

9.287687 

10.64 

9.991674 

9.296013 

11.06 

10.703987 

49 

12 

8326 

10.63 

649 

6677 

11.04 

33*3 

48 

13 

8964 

10.61 

624 

7339 

11.03 

2661 

47 

14 
15 

9.289600 
9.290236 

10.59 
10.58 

599 
574 

8001 
8662 

II.OI 
11.00 

1999 
1338 

46 
45 

16 

0870 

10.56 

549 

9322 

10.98 

0678 

44 

17 

1504 

10.54 

524 

9.299980 

10.96 

10.700020 

43 

18 

2137 

498 

9.300638 

10.95 

10.699362 

42 

19 
20 

2768 
3399 

10.51 
10.50 

£3 

1295 
1951 

10.93 

10.92 

8705 
8049 

41 

40 

21 
22 

9.294029 

4658 

10.48 
10.46 

9.991422 

397 

.42 

9.302607 
3261 

10.90 
10.89 

10.697393 
6739 

39 
38 

23 

5286 

10.45 

•43 

39H 

10.87 

6086 

37 

24 

5913 

10.43 

346 

4567 

10.86 

5433 

36 

25 

6539 

10.42 

321 

5218 

10.84 

4782 

35 

26 

7164 

10.40 

295 

5869 

10.83 

4131 

34 

27 

7788 

10.39 

270 

6519 

10.81 

3481 

33 

28 

8412 

10.37 

244 

7168 

10.80 

2832 

32 

29 

9034 

10.36 

2i8 

7815 

10.78 

2185 

31 

30 

9.299655 

10.34 

193 

8463 

10.77 

1537 

30 

31 
32 

9.300276 
0895 

10.32 
10.31 

9.991167 
141 

9109 
9-309754 

10.75 
10.74 

0891 
10.690246 

29 

28 

33 

1514 

10.29 

"5 

9.310398 

10.73 

10.689602 

27 

34 

2132 

10.28 

090 

1042 

10.71 

8958 

26 

35 

2748 

10.26 

064 

1685 

10.70 

8315 

25 

36 

3364 

10.25 

038 

2327 

10.68 

7673 

24 

37 

3979 

10.23 

9.991012 

2967 

10.67 

7033 

23 

38 

4593 

10.22 

9.990986 

3608 

10.65 

6392 

22 

39 
40 

5207 
5819 

10.20 
10.19 

960 
934 

•43 
•44 

4247 
4885 

10.64 
10.62 

5753 

21 
20 

41 

9.306430 

10.17 

9.990908 

9.315523 

10.61 

10.684477 

19 

42 

7041 

16.10 

882 

6159 

10.60 

3841 

18 

43 

7650 

10.14 

855 

6795 

10.58 

3205 

17 

44 

8259 

10.13 

829 

7430 

10.57 

1570 

16 

45 

8867 

I  O.I  I 

803 

8064 

10.55 

1936 

15 

46 

9.309474 

10.10 

777 

8697 

10.54 

1303 

14 

47 

9.310080 

1  0.08 

75° 

9329 

10.53 

0671 

13 

48 

0685 

10.07 

724 

9.319961 

10.51 

10.680039 

12 

49 

1289 

1  0.06 

697 

9.320592 

10.50 

10.679408 

11 

50 

1893 

10.04 

671 

1222 

10.48 

8778 

10 

51 

9-312495 

10.03 

9.990644 

9.321851 

10.47 

10.678149 

9 

52 

3097 

IO.OI 

618 

2479 

10.45 

7521 

8 

53 

3698 

10.00 

591 

3106 

10.44 

6894 

7 

54 

4297 

9.98 

565 

3733 

10.43 

6267 

6 

55 

4897 

9-97 

538 

•44 

4358 

10.41 

5642 

5 

56 

5495 

9.96 

5" 

•45 

4983 

10.40 

5017 

4 

57 

6092 

9-94 

485 

5607 

10.39 

4393 

3 

58 

6689 

9-93 

458 

6231 

10.37 

3769 

2 

,  59 

7284 

9.91 

43  * 

•45 

6853 

10.36 

3*47 

1 

60 

9.317879 

9.990404 

9.327475 

10.672525 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

101°                                     78°  ! 

25 


53 


12°            XiOCtA 

167° 

RXTXXIMiXG 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.  V 

Tang. 

Diff.  1" 

Cotang. 

0 

9.317879 

9.90 

9.990404 

•45 

9.327474 

10.35 

10.672526 

60 

1 
2 

9066 

9.88 
9.87 

378 
351 

8095 
8715 

io-33 
10.32 

1905 
1285 

59 

58 

3 

9.319658 

9.86 

3*4 

9334 

10.30 

0666 

57 

4 

9.320249 

9-84 

297 

9-3*9953 

10.29 

10.670047 

56 

5 

0840 

9.83 

270 

9-330570 

10.28 

10.669430 

55 

6 

1430 

9.82 

243 

1187 

10.26 

8813 

54 

7 

2019 

9.80 

215 

1803 

10.25 

8197 

53 

8 

2607 

9-79 

188 

2418 

10.24 

7582 

52 

9 

3'94 

9-77 

161 

3°33 

10.23 

6967 

51 

10 

3780 

9.76 

134 

•45 

3646 

10.21 

6354 

50 

11 

9.324366 

9-75 

9.990107 

.46 

9-334*59 

IO.2O 

10.665741 

49 

12 

4950 

9-73 

079 

4871 

IO.I9 

5129 

48 

13 

5534 

9-7* 

052 

5482 

10.17 

4518 

47 

14 

6117 

9.70 

9.990025 

6093 

10.16 

39°7 

46 

15 

6700 

9.69 

9.989997 

6702 

10.15 

3*98 

45 

16 

7281 

9.68 

970 

73" 

10.13 

2689 

44 

17 

7862 

9.66 

94* 

7919 

10.12 

2081 

43 

18 

8442 

9.65 

915 

85*7 

I  O.I  I 

1473 

42 

19 

9021 

9.64 

887 

9133 

IO.IO 

0867 

41 

20 

9-3*9599 

9.62 

860 

9-339739 

1  0.08 

10.660261 

40 

21 

9.330176 

9.61 

9.989832 

9-340344 

10.07 

10.659656 

39 

22 

0753 

9.60 

804 

0948 

1  0.06 

9052 

38 

23 

1329 

9-58 

111 

.46 

1552 

10.04 

8448 

37 

24 

1901 

9-57 

749 

•47 

2155 

10.03 

7845 

36 

25 

2478 

721 

*757 

10.02 

7*43 

35 

26 

3051 

9-54 

693 

3358 

10.01 

6642 

34 

27 

3624 

9-53 

665 

3958 

9-99 

6042 

33 

28 

4J95 

9-5* 

637 

4558 

9-98 

544* 

32 

29 

4766 

9.50 

609 

5157 

9-97 

4843 

31 

30 

5337 

9-49 

582 

5755 

9.96 

4*45 

30 

31 

9.335906 

9.48 

9-989553 

9-346353 

9-94 

10.653647 

29 

32 

6475 

9.46 

5*5 

6949 

9-93 

3051 

28 

33 

7043 

9-45 

497 

7545 

9.92 

27 

34 

7610 

9-44 

469 

8141 

9.91 

1859 

26 

35 

8176 

9-43 

44  1 

8735 

9.90 

1265 

25 

36 

8742 

9.41 

413 

93*9 

9.88 

0671 

24 

37 

9306 

9.40 

384 

9.349922 

9.87 

10.650078 

23 

38 

9.339871 

9-39 

356 

9-3505I4 

9.86 

10.649486 

22 

39 

9.340434 

9-37 

328 

1106 

9-85 

8894 

21 

40 

0996 

9-36 

300 

1697 

9-83 

8303 

20 

41 

9-34I558 

9-35 

9.989271 

9.352287 

9.82 

10.647713 

19 

42 

2119 

9-34 

243 

2876 

9.81 

7124 

18 

43 

2679 

9-3* 

•*4 

3465 

9.80 

6535 

17 

44 

3*39 

9-31 

186 

4053 

9-79 

5947 

16 

45 

3797 

9-3° 

157 

•47 

4640 

9-77 

15 

46 

4355 

9.29 

128 

.48 

5227 

9.76 

4773 

14 

47 

4912 

9.27 

100 

5813 

9-75 

4187 

13 

48 

5469 

9.26 

071 

6398 

9-74 

3602 

12 

49 

6024 

9.25 

042 

6982 

9-73 

3018 

11 

50 

6579 

9.24 

9.989014 

7566 

9.71 

2434 

10 

51 

9-347I34 

9.22 

9.988985 

9.358149 

9.70 

10.641851 

9~ 

52 

7687 

9.21 

956 

8731 

9.69 

1269 

8 

53 
54 

8240 
8792 

9.20 
9-  '9 

9*7 

9313 
9-359893 

9.68 
9.67 

0687 
10.640107 

7 
6 

55 

9343 

9.17 

869 

9.360474 

9.66 

10.639526 

5 

56 

9-349893 

9.16 

840 

.48 

1053 

9.65 

8947 

4 

57 

9.350443 

9.15 

811 

•49 

1632 

9.63 

8368 

3 

58 

0992 

9.14 

782 

•49 

2210 

9.62 

7790 

2 

59 

154° 

9-J3 

753 

•49 

2787 

9.61 

7213 

1 

60 

9.352088 

9.988724 

9.363364 

10.636636 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

102°                                      77° 

13°        SINES  AND  TANGENTS.      166° 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.352088 

9.11 

9.988724 

•49 

9.363364 

9.60 

10.636636 

60 

1 

2635 

9.10 

8695 

394° 

9-59 

6060 

59 

2 

3181 

9.09 

8666 

4515 

9-58 

5485 

58 

3 

3726 

9.08 

8636 

5090 

9-57 

4910 

57 

4 
5 

4271 
4815 

9.07 
9.05 

8607 
8578 

5664 
6237 

9-55 
9-54 

4336 
3763 

56 
55 

6 

5358 

9.04 

8548 

6810 

9-53 

3190 

54 

7 

5901 

9.03 

8519 

7382 

9.52 

2618 

53 

8 

6443 

9.02 

8489 

7953 

9.51 

2047 

52 

9 

6984 

9.01 

8460 

8524 

9-5° 

1476 

51 

10 

75*4 

8.99 

8430 

9094 

9-49 

0906 

50 

11 
12 

9.358064 
8603 

8.98 
8.97 

9.988401 
8371 

9.369663 
9.370232 

9.48 
9.46 

10.630337 
10.629768 

49 

48 

13 

9141 

8.96 

8342 

•49 

0799 

9-45 

9201 

47 

14 

9.359678 

8.95 

8312 

•5° 

1367 

9-44 

8633 

46 

15 

9.360215 

8.93 

8282 

1933 

9-43 

8067 

45 

16 

0752 

8.92 

8252 

2499 

9.42 

7501 

44 

17 

1287 

8.91 

8223 

3064 

9.41 

6936 

43 

18 

1822 

8.90 

8193 

3629 

9.40 

6371 

42 

19 

2356 

8.89 

8163 

4J93 

9-39 

5807 

41 

20 

2889 

8.88 

8133 

4756 

9.38 

5*44 

40 

21 

9.363422 

8.87 

9.988103 

9-3753I9 

9-37 

10.624681 

39 

22 

3954 

8.85 

8073 

5881 

9-35 

4119 

38 

23 

4485 

8.84 

8043 

6442 

9-34 

3558 

37 

24 

5016 

8.83 

8013 

7003 

9-33 

2997 

36 

25 

5546 

8.82 

7983 

7563 

9.32 

2437 

35 

26 

6075 

8.81 

7953 

8122 

9.31 

1878 

34 

27 

6604 

8.80 

7922 

8681 

9.30 

1319 

33 

28 

7131 

8.79 

7892 

9*39 

9.29 

0761 

32 

29 

7659 

8.78 

7862 

•5° 

9797 

9.28 

0203 

31 

30 

8185 

8.76 

7832 

9-380354 

9.27 

10.619646 

30 

31 

9.368711 

8.75 

9.987801 

9.380910 

9.26 

10.619090 

29 

32 

9236 

8.74 

7771 

1466 

9-*5 

8534 

28 

33 

9.369761 

8.73 

7740 

2020 

9.24 

7980 

27 

34 

9.370285 

8.72 

7710 

2575 

9-*3 

74*5 

26 

35 

0808 

8.71 

7679 

3129 

9.22 

6871 

25 

36 

J33° 

8.70 

7649 

3682 

9.21 

6318 

24 

37 

1852 

8.69 

7618 

4*34 

9.20 

5766 

23 

38 

*373 

8.67 

7588 

4786 

9.19 

52I4 

22 

39 

2894 

8.66 

7557 

9.18 

4663 

21 

40 

34H 

8.65 

75*6- 

5888 

9.17 

4112 

20 

41 

9-373933 

8.64 

9.987496 

9.386438 

9.15 

10.613562 

19 

42 

445* 

8.63 

7465 

6987 

9.14 

3013 

18 

43 

4970 

8.62 

7434 

•51 

7536 

9-1  3 

2464 

17 

44 

5487 

8.61 

7403 

•5* 

8084. 

9.12 

1916 

16 

45 

6003 

8.60 

737* 

8631 

9.11 

1369 

15 

46 

6519 

8.59 

7341 

9178 

9.10 

0822 

14 

47 

7035 

8.58 

7310 

9.389724 

9.09 

10.610276 

13 

48 

7549 

8.57 

7*79 

9.390270 

9.08 

10.609730 

12 

49 

8063 

8.56 

7248 

0815 

9.07 

9185 

11 

50 

8577 

8-54 

7217 

1360 

9.06 

8640 

10 

51 

9089 

8-53 

9.987186 

9.391903 

9-°5 

10.608097 

9 

52 
53 

9.379601 
9.380113 

8.52 
8.51 

7155 
7124 

2989 

9.04 
9.03 

7553 
7011 

8 

7 

54 

0624 

8.50 

7092 

9.02 

6469 

6 

55 

1134 

8.49 

7061 

4°73 

9.01 

59*7 

5 

56 

1643 

8.48 

7030 

4614 

9.00 

5386 

4 

57 

58 

2152 
2661 

8.47 
8.46 

6998 
6967 

5J54 
5694 

8.'98 

4846 
4306 

3 

2 

59 
60 

3168 
9-383675 

8.45 

6936 
9.986904 

•5* 

6*33 
9.396771 

8.97 

3767 
10.603229 

1 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l' 

Cotang.  I  Diff.l" 

Tang. 

M. 

103°                                     76° 

55 


14°            LOGAR.ITHIYIIC           165° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  I" 

Cotang. 

0 

9.383675 

8.44 

9.986904 

.52 

9.396771 

8.96 

10.603229 

60 

1 

4182 

8.43 

6873 

•53 

73°9 

8.96 

2691 

59 

2 

4687 

8.42 

6841 

7846 

8-95 

2154 

58 

3 

5*92 

8.41 

6809 

8383 

8.94 

1617 

57 

4 

5697 

8.40 

6778 

8919 

8-93 

1081 

56 

5 

6201 

8-39 

6746 

9455 

8.92 

°545 

55 

6 

6704 

8.38 

6714 

9-39999° 

8.91 

10.600010 

54 

7 

7207 

8.37 

6683 

9.400524 

8.90 

iQ-599476 

53 

8 

7709 

8.36 

6651 

1058 

8.89 

8942 

52 

9 

8210 

8-35 

6619 

1591 

8.88 

8409 

51 

10 

8711 

8.34 

6587 

2124 

8.87 

7876 

50 

11 

9211 

8.33 

9-986555 

9.402656 

8.86 

10.597344 

49 

12 

9.389711 

8.32 

6523 

3187 

8.85 

6813 

48 

13 

9.390210 

8.31 

6491 

37i8 

8.84 

6282 

47 

14 

0708 

8.30 

6459 

4249 

8.83 

5751 

46 

15 

1206 

8.28 

6427 

4778 

8.82 

5222 

45 

16 

1703 

8.27 

6395 

•53 

5308 

8.81 

4692 

44 

17 

2199 

8.26 

6363 

•54 

5836 

8.80 

4164 

43 

18 

2695 

8.25 

633J 

6364 

8.79 

3636 

42 

19 

V91 

8.24 

6299 

6892 

8.78 

3108 

41 

20 

3685 

8.23 

6266 

7419 

8.77 

2581 

40 

21 

9-394I79 

8.22 

9.986234 

9.407945 

8.76 

10.592055 

39 

22 
23 

$1 

8.21 
8.20 

6202 
6169 

8471 
8997 

8-75 

8.74 

1529 
1003 

38 
37 

24 

5658 

8.20 

6137 

9.409521 

8.74 

10.590479 

36 

25 

6150 

8.18 

6104 

9.410045 

8-73 

io-589955 

35 

26 

6641 

8.17 

6072 

0569 

8.72 

943  1 

34 

27 

7132 

8.17 

6039 

1092 

8.71 

8908 

33 

28 

7621 

8.16 

6007 

1615 

8.70 

8385 

32 

29 

Sin 

8.15 

5974 

2137 

8.69 

7863 

31 

30 

8600 

8.14 

5942 

•54 

2658 

8,68 

734^ 

30 

31 

9088 

8.13 

9.985909 

•55 

9-4I3I79 

8.67 

10.586821 

29 

32 

9-399575 

8.12 

5876 

3699 

8.66 

6301 

28 

33 

9.400062 

8.ii 

5843 

4219 

8.65 

5781 

27 

34 

0549 

8.10 

5811 

4738 

8.64 

5262 

26 

35 

i°35 

8.09 

5778 

5*57 

8.64 

4743 

25 

36 

1520 

8.08 

5745 

5775 

8.63 

4225 

24 

37 

2005 

8.07 

57^ 

6293 

8.62 

37°7 

23 

38 

2489 

8.06 

5679 

6810 

8.61 

3190 

22 

39 

2972 

8.05 

5646 

7326 

8.60 

2674 

21 

40 

3455 

8.04 

56i3 

7842 

8.59 

2158 

20 

41 

9.403938 

8.03 

9.985580 

9-418358 

8.58 

10.581642 

19 

42 

4420 

8.02 

5547 

8873 

8.57 

1127 

18 

43 

4901 

8.01 

55H 

9387 

8.56 

0613 

17 

44 

5382 

8.00 

.   5480 

9.419901 

8.55 

10.580099 

16 

45 

5862 

7-99 

5447 

•55 

9.420415 

8.55 

i°-579585 

15 

46 

6341 

7.98 

54H 

.56 

0927 

8.54 

9073 

14 

47 

6820 

7-97 

538o 

1440 

8-53 

8560 

13 

48 

7299 

7.96 

5347 

1952 

8.52 

8048 

12 

49 

7777 

7-95 

53H 

2463 

8.51 

7537 

11 

50 

8254 

7-94 

5280 

2974 

8.50 

7026 

10 

51 

9.408731 

7-94 

9.985247 

9.423484 

8-49 

10.576516 

9 

52 

9207 

7-93 

5*13 

3993 

8.48 

6007 

8 

53 

9.409682 

7.92 

5180 

45°3 

8.48 

5497 

7 

54 

9.410157 

7.91 

5146 

5011 

8.47 

4989 

6 

55 

0632 

7.90 

5"3 

5519 

8.46 

4481 

5 

56 

1106 

7.89 

5079 

6027 

8.45 

3973 

4 

57 

1579 

7.88 

5°45 

6534 

8.44 

3466 

3 

58 

2052 

7.87 

5011 

7041 

8.43 

2959 

2 

59 

2524 

7.86 

4978 

•56 

7547 

8.43 

2453 

1 

60 

9.412996 

9-984944 

9.428052 

10.571948 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

104°                                      75° 

15°       SINES  AND  TANGENTS.       164°  : 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.412996 

7-85 

9.984944 

•57 

9.428052 

8.42 

10.571948 

60 

1 

3467 

7-84 

4910 

8557 

8.41 

1443 

59 

2 

393s 

7.83 

4876 

9062 

8.40 

0938 

58 

3 

4408 

7.83 

4842 

9.429566 

8-39 

10.570434 

57 

4 

4878 

7.82 

4808 

9.430070 

8.38 

10.569930 

56 

5 

5347 

7.81 

4774 

0573 

8.38 

94*7 

55 

6 

5815 

7.80 

474° 

1075 

8-37 

8925 

54 

7 

6283 

7-79 

4706 

1577 

8.36 

8423 

53 

8 

6751 

7-78 

4672 

2079 

8-35 

7921 

52 

9 

10 

7217 
7684 

7-77 
7.76 

4637 
4603 

2580 
3080 

8.34 
8-33 

7420 
6920 

51 
50 

Tf 

9.418150 

7-75 

9.984569 

9.433580 

8.32 

10.566420 

49 

12 

8615 

7-74 

4535 

4080 

8.32 

59*o 

48 

13 

9079 

7-73 

4500 

4579 

8.31 

54*i 

47 

14 

9.419544 

7-73 

4466 

•57 

5°78 

8.30 

4922 

46 

15 

9.420007 

7.72 

4432 

•58 

5576 

8.29 

44*4 

45 

16 

0470 

7.71 

4397 

6073 

8.28 

39*7 

44 

17 

°933 

7.70 

4363 

6570 

8.28 

343° 

43 

18 

1395 

7.69 

43*8 

7067 

8.27 

*933 

42 

19 

1857 

7.68 

4294 

7563 

8.26 

*437 

41 

20 

2318 

7.67 

4*59 

8059 

8.25 

1941 

40 

21 

9.422778 

7.67 

9.984224 

9-438554 

8.24 

10.561446 

~39~ 

22 

3238 

7.66 

4190 

9048 

8.23 

0952 

38 

23 

3697 

7.65 

4155 

9-439543 

8.23 

10.560457 

37 

24 

4156 

7-64 

4120 

9.440036 

8.22 

10.559964 

36 

25 

4615 

7.63 

4085 

0529 

8.21 

35 

26 

5°73 

7.62 

4050 

1022 

S,20 

8978 

34 

27 
28 

5530 
5987 

7.61 
7.60 

4015 
398i 

2006 

b.,9, 
8.19 

8486 
7994 

33 

32 

29 

6443 

7.60 

3946 

2497 

8.18 

75°3 

31 

30 

6899 

7-59 

3911 

2988 

8.17 

7012 

30 

31 

9-4*7354 

7.58 

9.983875 

•58 

9-443479 

8.16 

10.556521 

29 

32 

7809 

7-57 

3840 

•59 

3968 

8.16 

6032 

28 

33 

8263 

7.56 

4458 

8.15 

554* 

27 

34 

8717 

7-55 

377° 

4947 

8.14 

5°53 

26 

35 

9170 

7-54 

3735 

5435 

8.13 

4565 

25 

36 

9.429623 

7-54 

3700 

59*3 

8.12 

4077 

24 

37 

9.430075 

7-53 

3664 

6411 

8.12 

3589 

23 

38 

0527 

7.52 

3629 

6898 

8.  ii 

3102 

22 

39 

0978 

3594 

7384 

8.10 

2616 

21 

40 

1429 

7.50 

3558 

7870 

8.09 

2130 

20 

41 

9.431879 

7-49 

9-9835*3 

9.448356 

8.09 

10.551644 

19 

42 

2329 

7-49 

3487 

8841 

8.08 

1159 

18 

43 

2778 

7.48 

345* 

9326 

8.07 

0674 

17 

44 

3226 

7-47 

34l6 

9.449810 

8.06 

10.550190 

16 

45 

3675 

7.46 

9.450294 

8.06 

10.549706 

15 

46 

4122 

7-45 

3345 

0777 

8.05 

9223 

14 

47 

4569 

7-44 

33°9 

•59 

1260 

8.04 

8740 

13 

48 

5016 

7-44 

3*73 

.60 

1743 

8.03 

8*57 

12 

49 

5462 

7-43 

3*38 

2225 

8.02 

7775 

11 

50 

5908 

7.42 

3202 

2706 

8.02 

7*94 

10 

51 

9-436353 

7.41 

9.983166 

9.453187 

8.01 

10.546813 

9 

52 

6798 

7.40 

3130 

3668 

8.00 

6332 

8 

53 

7242 

7.40 

3°94 

4148 

7-99 

585* 

7 

54 

7686 

7-39 

3058 

4628 

7-99 

537* 

6 

55 

8129 

7.38 

3022 

5107 

7.98 

4893 

5 

56 

857* 

7-37 

2986 

5586 

7-97 

4414 

4 

57 

9014 

7.36 

2950 

6064 

7.96 

3936 

3 

58 

9456 

7.36 

2914 

6542 

7.96 

3458 

2 

59 

9-439897 

7-35 

2878 

.60 

7019 

7-95 

2981 

1 

60 

9.440338 

9.982842 

9.457496 

10.542504 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang.    Diff.  1" 

Tang. 

M. 

105°                                     74° 

57 


16°            LOGARITHMIC           163° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang.   j  Diff.  1" 

Cotang. 

0 

9-440338 

7-34 

9.982842 

.60 

9.457496 

7-94 

10.542504 

60 

1 

0778 

7-33 

2805 

.60 

7973 

7-93 

2027 

59 

2 

1218 

7.32 

2769 

.61 

8449 

7-93 

1551 

58 

3 
4 

1658 
2096 

7-3i 

2733 
2696 

8925 

9400 

7.92 
7.91 

1075 
0600 

57 
56 

5 

2535 

7-3° 

2660 

9.459875 

7.90 

10.540125 

55 

6 

2973 

7.29 

2624 

9.460349 

7.90 

10.539651 

54 

7 

3410 

7.28 

2587 

0823 

7.89 

9177 

53 

8 

3847 

7.27 

2551 

1297 

7.88 

8703 

52 

9 

4284 

7.27 

2514 

1770 

7.88 

8230 

51 

10 

4720 

7.26 

2477 

2242 

7.87 

7758 

50 

TT 

9-445  *  55 

7.25 

9.982441 

9.462714 

7.86 

10.537286 

49 

12 

559° 

7.24 

2404 

3186 

7.85 

6814 

48 

13 

6025 

2367 

3658 

7.85 

6342 

47 

14 

6459 

7-23 

2331 

4129 

7.84 

587J 

46 

15 

6893 

7.22 

2294 

4599 

7.83 

45 

16 

7326 

7.21 

2257 

.61 

5069 

7-83 

493  1 

44 

17 

7759 

7.20 

2220 

.62 

5539 

7.82 

4461 

43 

18 

8191 

7.20 

2183 

6008 

7.81 

3992 

42 

19 

8623 

7.19 

2146 

6476 

7.80 

3524 

41 

20 

9°54 

7.18 

2109 

6945 

7.80 

3055 

40 

21 

9485 

7.17 

9.982072 

9.467413 

7-79 

10.532587 

39 

22 

9-4499J5 

7.16 

2035 

7880 

7.78 

2I2O 

38 

23 

9-450345 

7.16 

I998 

8347 

7.78 

1653 

37 

24 

0775 

7.15 

1961 

8814 

7-77 

1186 

36 

25 

1204 

7.14 

1924 

9280 

7.76 

0720 

35 

26 

1632 

7.13 

1886 

9.469746 

7-75 

10.530254 

34 

27 

28 

2060 
2488 

7.12 

l849 

1812 

9.470211 
0676 

7-75 
7-74 

10.529789 
9324 

33 
32 

29 

2915 

7.11 

1774 

1141 

7-73 

8859 

31  1 

30 

3342 

7.10 

1737 

.62 

1605 

7-73 

8395 

30 

31 

9.453768 

7.10 

9.981699 

.63 

9.472068 

7.72 

10.527932 

29 

32 

4194 

7-09 

l662 

2532 

7.71 

7468 

28 

33 

4619 

7.08 

1625 

2995 

7.71 

7005 

27 

34 

5°44 

7.07 

1587 

3457 

7.70 

6543 

26 

35 

5469 

7.07 

1549 

39*9 

7.69 

6081 

25 

36 

5893 

7.06 

1512 

4381 

7.69 

5619 

24 

37 

6316 

7.05 

1474 

4842 

7.68 

5*58 

23 

38 

6.739 

7.04 

1436 

53°3 

7.67 

4697 

22 

39 

7162 

7.04 

1399 

5763 

7.67 

4237 

21 

40 

7584 

7.03 

1361 

6223 

7.66 

3777 

20 

41 

9.458006 

7.02 

9.981323 

9-476683 

7.65 

10.523317 

19 

42 

8427 

7.01 

1285 

7142 

7.65 

2858 

18 

43 

8848 

7.01 

1247 

7601 

7.64 

2399 

17 

44 

9268 

7.00 

1209 

8059 

7.63 

1941 

16 

45 

9.459688 

6.99 

II7I 

.63 

8517 

7-63 

1483 

15 

46 

9.46010? 

6.98 

"33 

.64 

8975 

7.62 

1025 

14 

47 

0527 

6.98 

1095 

9432 

7.61 

0568 

13 

48 

0946 

6.97 

1057 

9.479889 

7.61 

10.520111 

12 

49 
50 

1364 
1782 

6.96 
6.95 

1019 

0981 

9.480345 
0801 

7.60 
7-59 

10.519655 
9199 

11 

10 

51 

9.462199 

6.95 

9.980942 

9.481257 

7-5S 

10.518743 

9 

52 

2616 

6.94 

0904 

1712 

8288 

8 

53 

3032 

6.93 

0866 

2167 

7-57 

7833 

7 

54 

3448 

6.93 

0827 

2621 

7-57 

7379 

6 

55 

3864 

6.92 

0789 

3°75 

7.56 

6925 

5 

56 

4279 

6.91 

0750 

3529 

7-55 

6471 

4 

57 

4694 

6.90 

0712 

3982 

7-55 

6018 

3 

58 

5108 

6.90 

067^ 

4435 

7-54 

5565 

2 

59 

5522 

6.89 

0631 

.64 

4887 

7-53 

1 

60 

9-465935 

9.980596 

9.485339 

10.514661 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l' 

Cotang. 

Diff.  1" 

Tang. 

M. 

106°                                     73° 

17°        SINES  AND  TANGENTS.       162° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  I" 

Cotang. 

0 

9-465935 

6.88 

9.980596 

.64 

9485339 

7-53 

10.514661 

60 

1 

6348 

6-88 

0558 

.64 

5791 

7-52 

4209 

59 

2 

6761 

6.87 

0519 

.65 

6242 

3758 

58 

3 

7173 

6.86 

0480 

6693 

7-51 

33°7 

57 

4 

5 

7585 
7996 

6.85 
6.85 

0442 
0403 

7H3 
7593 

7.50 
7-49 

2857 
2407 

56 
55 

6 

8407 

6.84 

0364 

8043 

7-49 

'957 

54 

7 

8817 

6.83 

0325 

8492 

7.48 

1508 

53 

8 

9227 

6.83 

0286 

8941 

7-47 

1059 

52 

9 

9.469637 

6.82 

0247 

939° 

7-47 

0610 

51 

10 

9.470046 

6.81 

0208 

9.489838 

7.46 

10.510162 

50 

11 

0455 

6.80 

9.980169 

9.490286 

7.46 

10.509714 

49 

12 

0863 

6.80 

0130 

°733 

7-45 

9267 

48 

13 

1271 

6.79 

0091 

1180 

7-44 

8820 

47 

14 

1679 

6.78 

0052 

1627 

7-44 

8373 

46 

15 

2086 

6.78 

9.980012 

2073 

7-43 

7927 

45 

16 

2492 

6.77 

9.979973 

.65 

2519 

7-43 

7481 

44 

17 

2898 

6.76 

9934 

.66 

2965 

7.42 

7°35 

43 

18 

33°4 

6.76 

9895 

3410 

7.41 

6590 

42 

19 

3710 

6.75 

9855 

3854 

7.40 

6146 

41 

20 

4115 

6.74 

9816 

4299 

7.40 

5701 

40 

21 

9.474519 

6.74 

9.979776 

9.494743 

7-39 

10.505257 

39 

22 

49*3 

6-73 

9737 

5186 

7-39 

4814 

38 

23 
24 

53*7 
573° 

6.72 
6.72 

9697 
9658 

5630 
6073 

7-38 
7-37 

4370 
3927 

37 
36 

25 

6133 

6.71 

9618 

6515 

7-37 

3485 

35 

26 

6536 

6.70 

9579 

6957 

7.36 

3°43 

34 

27 

6938 

6.69 

9539 

7399 

7.36 

2601 

33 

28 

7340 

6.69 

9499 

7841 

7-35 

2159 

32 

29 

7741 

6.68 

9459 

8282 

7-34 

1718 

31 

30 

8142 

6.67 

9420 

8722 

7-34 

1278 

30 

32 

9.478542 
8942 

6.67 
6.66 

9.979380 
934° 

.66 

9161 
9.499603 

7-33 
7-33 

0837 
10.500397 

29 

28 

33 

9342 

6.65 

9300 

.67 

9.500042 

7-3* 

10.499958 

27 

34 

9.479741 

6.65 

9260 

0481 

7-31 

26 

35 

9.480140 

6.64 

9220 

0920 

7-31 

9080 

25 

36 

0539 

6.63 

9180 

1359 

7-3° 

8641 

24 

37 

0937 

6.63 

9140 

1797 

7.30 

8203 

23 

38 

133^ 

6.62 

9100 

2235 

7.29 

7765 

22 

39 

6.61 

9059 

2672 

7.28 

7328 

21 

40 

2128 

6.61 

9019 

3ioc 

7.28 

6891 

20 

41 

9.482525 

6.60 

9.978979 

9.503546 

7.27 

10.496454 

19 

42 

2921 

6.59 

8939 

3982 

7.27 

601  8 

18 

43 

3316 

6.59 

8898 

4418 

7.26 

5582 

17 

44 

3712 

6.58 

8858 

4854 

7.25 

5146 

16 

45 

4107 

6-57 

8817 

5289 

7-^5 

4711 

15 

46 

4501 

6.57 

8777 

57*4 

7.24 

4276 

14 

47 

4895 

6.56 

8736 

.67 

6159 

7.24 

3841 

13 

48 

5289 

6-55 

8696 

.68 

6593 

7.23 

34°7 

12 

49 

5682 

6.55 

8655 

7027 

7.22 

2973 

11 

50 

0075 

6.54 

8615 

7460 

7.22 

2540 

10 

61 

9.486467 

6-53 

9.978574 

9.507893 

7.21 

10.492107 

9 

52 

6860 

6.53 

8533 

8326 

7.21 

1674 

8 

53 

7251 

6.52 

8493 

8759 

7.20 

1241 

hr 
t 

54 

7643 

6,51 

8452 

9191 

7.19 

0809 

6 

55 

803. 

6.51 

8411 

9.509622 

7.19 

10.490378 

5 

56 
57 

8424 
8814 

6.50 
6.50 

8370 
8329 

9.51005: 
0485 

7.18 
7.18 

10.489946 
9515 

4 
3 

58 

920^ 

6.49 

8288 

0911 

7.17 

9084 

2 

59 

9593 

6.48 

8247 

.68 

1346 

7.17 

8654 

1 

60 

9.489982 

9.978201 

9.511776 

10.488224 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l' 

Cotang. 

Diff.  1" 

Tang. 

M. 

107°                                     72° 

59 


J.O                  JLrv  vr^*ix  J.  i  iiJLVJLXv/                J.O1 

M. 

Sine. 

Diff.  V 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 
1 

9.489982 
9.490371 

6.48 

6.47 

9.978206 
8165 

.68 

9.511776 
2206 

7.16 
7.l6 

10.488224 
7794 

60 
59 

2 

0759 

6.46 

8124 

.68 

2635 

7-15 

7365 

58 

3 

1147 

6.46 

8083 

.69 

3064 

7.14 

6936 

57 

4 

1535 

6-45 

8042 

3493 

7.14 

6507 

56 

5 

1922 

6.44 

8001 

3921 

7-13 

6079 

55 

6 

2308 

6.44 

7959 

4349 

7-13 

565! 

54 

7 

2695 

6.43 

7918 

4777 

7.12 

5223 

53 

8 

3081 

6.42 

7877 

5204 

7-12 

4796 

52 

9 

3466 

6.42 

7835 

563' 

7-II 

4369 

51 

10 

3851 

6.41 

7794 

6057 

7.10 

3943 

50 

11 

9.494236 

6.4I 

9.977752 

9.516484 

7.10 

10.483516 

49 

12 

4621 

6.40 

77" 

6910 

7.09 

3090 

48 

13 
14 

5°°5 
5388 

6-39 
6.39 

7669 
7628 

7335 
7761 

7.09 
7.08 

2665 
2239 

47 
46 

15 

577* 

6.38 

7586 

.69 

8185 

7.08 

1815 

45 

16 

6154 

6.37 

7544 

.70 

8610 

7.07 

1390 

44 

17 

6537 

6.37 

75°3 

9034 

7.06 

0966 

43 

18 

6919 

6.36 

7461 

9458 

7.06 

0542 

42 

19 

7301 

6.36 

74*9 

9.519882 

7.05 

10.480118 

41 

20 

7682 

6.35 

7377 

9.520305 

7.05 

10.479695 

40 

~21 

9.498064 

6.34 

9-977335 

0728 

7.04 

9272 

39 

22 

8444 

6.34 

7*93 

"5i 

7.04 

8849 

38 

23 

8825 

6.33 

7251 

1573 

7.03 

8427 

37 

24 

9204 

6.32 

7209 

1995 

7-°3 

8005 

36 

25 

9584 

6.32 

7167 

2417 

7.02 

7583 

35 

26 

9.499963 

6.31 

7125 

2838 

7.02 

7162 

34 

27 
28 

9.500342 
0721 

6.31 
6.30 

7083 
7041 

3*59 
3680 

7-01 
7.01 

6741 
6320 

33 

32 

29 

1099 

6.29 

6999 

4100 

7.00 

5900 

31 

30 

1476 

6.29 

6957 

4520 

6.99 

548o 

30 

31 

9.501854 

6.28 

9.976914 

.70 

9-5*4939 

6.99 

10.475061 

29 

32 

2231 

6.28 

6872 

•7i 

5359 

6.98 

4641 

28 

33 
34 

2607 
2984 

6.27 
6.26 

6830 
6787 

5778 
6197 

6.98 
6.97 

4222 
3803 

27 
26 

35 

3360 

6.26 

6745 

6615 

6.97 

3385 

25 

36 

3735 

6.25 

6702 

7033 

6.96 

2967 

24 

37 

4110 

6.25 

6660 

745  « 

6.96 

*549 

23 

38 

4485 

6.24 

6617 

7868 

6.95 

2132 

22 

39 

4860 

6.23 

6574 

8285 

6-95 

1715 

21 

40 

5*34 

6.23 

6532 

8702 

6.94 

1298 

20 

41 

9.505608 

6.22 

9.976489 

9.529119 

6-93 

10.470881 

19 

42 

598i 

6.22 

6446 

9535 

6.93 

0465 

18 

43 

44 

6354 
6727 

6.21 
6.20 

6404 
6361 

9.529950 
9.530366 

6.93 
6.92 

10.470050 
10.469634 

17 
16 

45 

7099 

6.20 

6318 

0781 

6.91 

9219 

15 

46 

747i 

6.19 

6275 

•7i 

1196 

6.91 

8804 

14 

47 

7843 

6.19 

6232 

.72 

1611 

6.90 

8389 

13 

48 

8214 

6.18 

6189 

2025 

6.90 

7975 

12 

49 

8585 

6.18 

6146 

*439 

6.89 

7561 

11 

50 

8956 

6.17 

6103 

*853 

6.89 

7H7 

10 

ftl 

52 

9326 
9.509696 

6.16 
6.16 

9.976060 
6017 

9.533266 
3679 

6.88 
6.88 

10.466734 
6321 

9 
8 

53 

9.510065 

6.15 

5974 

4092 

6.87 

5908 

7 

54 

°434 

6.15 

593° 

4504 

6.87 

5496 

6 

55 

0803 

6.14 

5887 

4916 

6.86 

5084 

5 

56 

1172 

6.13 

5844 

53*8 

6.86 

4672 

4 

57 

1540 

6.13 

5800 

5739 

6.85 

4261 

3 

58 

1907 

6.12 

5757 

6150 

6.85 

3850 

2 

59 

2275 

6.12 

57H 

.72 

6561 

6.84 

3439 

1 

60 

9.512642 

9.975670 

9.536972 

10.463028 

0 

Cosine. 

Diff.  1"     Sine. 

Diff.l" 

Cotang. 

Diff.  V 

Tang. 

M. 

108°                                     71° 

60 


19°        SINES  AND  TANGENTS.       160° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.512642 

6.  ii 

9.975670 

•73 

9.536972 

6.84 

10.463028 

60 

1 

3009 

6.  ii 

5627 

7382 

6.83 

2618 

59 

2 

3375 

6.10 

5583 

779* 

6.83 

2208 

58 

3 

6.09 

5539 

8202 

6.82 

1798 

57 

4 

4107 

6.09 

5496 

8611 

6.82 

1389 

56 

5 

447* 

6.08 

545* 

9020 

6.81 

0980 

55 

6 

4837 

6.08 

5408 

94*9 

6.81 

0571 

54 

7 

5202 

6.07 

5365 

9-539837 

6.80 

10.460163 

53 

8 

5566 

6.07 

53*1 

9.540245 

6.80 

10-459755 

52 

9 
10 

593° 
6294 

6.06 
6.05 

5*77 
5*33 

0653 
1061 

6.79 
6.79 

9347 
8939 

51 

50 

11 

9.516657 

6.05 

9.975189 

9.541468 

6.78 

10.458532 

49 

12 

7020 

6.04 

5*45 

1875 

6.78 

8125 

48 

13 

7382 

6.04 

5101 

2281 

6.77 

7719 

47 

14 

7745 

6.03 

5°57 

2688 

6.77 

7312 

46 

15 

8107 

6.03 

5OI3 

•73 

3«>94 

6.76 

6906 

45 

16 

8468 

6.02 

4969 

•74 

3499 

6.76 

6501 

44 

17 

8829 

6.01 

49*5 

39°5 

6.75 

6095 

43 

18 

9190 

6.01 

4880 

4310 

6.75 

5690 

42 

19 

6.00 

4836 

4715 

6.74 

5*85 

41 

20 

9.519911 

6.00 

479* 

5"9 

6.74 

4881 

40 

21 

9.520271 

5-99 

9.974748 

9-5455*4 

6.73 

10.454476 

39 

22 
23 

0631 
0990 

5-99 
5.98 

47°3 
4659 

59*8 
6331 

6.73 
6.72 

3669 

38 
37 

24 

1349 

5.58 

4614 

6735 

6.72 

3265 

36 

25 

1707 

5-97 

457° 

7138 

6.71 

2862 

35 

26 

2066 

5-96 

45*5 

7540 

6.71 

2460 

34 

27 

2424 

5.96 

4481 

7943 

6.70 

2057 

33 

28 

2781 

5-95 

443  6 

8345 

6.70 

l655 

32 

29 

3138 

5-95 

439' 

•74 

8747 

6.69 

i*53 

31 

30 

3495 

5-94 

4347 

•75 

9149 

6.69 

0851 

30 

31 

9.523852 

5-94 

9.974302 

9550 

6.68 

0450 

29 

32 

4208 

5-93 

4*57 

9.549951 

6.68 

10.450049 

28 

33 

4564 

5-93 

4212 

9-55°35* 

6.67 

10.449648 

27 

34 

4920 

5-9* 

4167 

0752 

6.67 

9248 

26 

35 

5*75 

5.91 

4122 

1152 

6.66 

8848 

25 

36 

5630 

5-91 

4077 

1552 

6.66 

8448 

24 

37 

5984 

5.90 

4032 

1952 

6.65 

8048 

23 

38 

6339 

5-9° 

3987 

6.65 

7649 

22 

39 

6693 

5.89 

394* 

2750 

6.65 

7*5° 

21 

40 

7046 

5-89 

3897 

3H9 

6.64 

6851 

20 

41 

9.527400 

5.88 

9.973852 

9-553548 

6.64 

10.446452 

19 

42 

'  7753 

5.88 

3807 

3946 

6.63 

6054 

18 

43 

8105 

5-87 

3761 

•75 

4344 

6.63 

5656 

17 

44 

8458 

5-87 

3716 

.76 

4741 

6.62 

5*59 

16 

45 

8810 

5.86 

3671 

5'39 

6.62 

4861 

15 

46 

9161 

5.86 

3625 

5536 

6.61 

4464 

14 

47 

95*3 

5-85 

5933 

6.61 

4067 

13 

48 

9.529864 

5-85 

353^5 

6329 

6.60 

3671 

12 

-49 

9.530215 

5.84 

6725 

6.60 

3*75 

11 

50 

0565 

5-84 

3444 

7121 

6.59 

2879 

10 

51 

9-53°9I5 

5.83 

9-973398 

9-5575I7 

6.59 

10.442483 

9 

52 

1265 

5.82 

335* 

79X3 

6.59 

2087 

8 

53 

1614 

5.82 

33°7 

8308 

6.58 

1692 

7 

54 

1963 

5.81 

3261 

8702 

6.58 

1298 

6 

55 

2312 

5.81 

3**5 

9097 

6.57 

0903 

5 

56 

2661 

5.80 

3169 

949  * 

6.57 

0509 

4 

57 

3009 

5.80 

3I24 

9-559885 

6.56 

10.440115 

3 

58 

3357 

5-79 

3078 

.76 

9.560279 

6.56 

10.439721 

2 

59 

37°4 

5-79 

3032 

•77 

0673 

6.55 

93*7 

1 

60 

9.534052 

9.972986 

9.561066 

10.438934 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

109°                                      70° 

til 


20°            XiOaARITHlKXC           159° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff  1" 

Cotang. 

0 

9.534052 

5-78 

9.972986 

•77 

9.561066 

6.55 

10.438934 

60  1 

1 

4399 

2940 

H59 

6.54 

8541 

59 

2 

4745 

5-77 

2894 

1851 

6.54 

8149 

58 

3 

5092 

5-77 

2848 

2244 

6-53 

7756 

57 

4 

5438 

2802 

2636 

6.53 

7364 

56 

5 

5783 

£76 

*755 

3028 

6.53 

6972 

55 

6 

6129 

5-75 

2709 

3419 

6.52 

6581 

54 

7 

6474 

5-74 

2663 

3811 

6.52 

6189 

53 

8 

6818 

5-74 

2617 

4202 

6.51 

5798 

52 

9 

7163 

5-73 

2570 

459* 

6.51 

5408 

51 

10 

7507 

5-73 

2524 

4983 

6.50 

5017 

50 

11 

9-53785I 

5-7* 

9.972478 

•77 

9-565373 

6.50 

10.434627 

49 

12 
13 

8194 
8538 

5-7* 

*43* 
*385 

.78 

5763 
6l53 

6.49 
6.49 

4*37 
3847 

48 
47 

14 

8880 

5.71 

2338 

6542 

6.49 

3458 

46 

15 

9223 

5.70 

2291 

6932 

6.48 

3068 

45 

16 

9565 

5-7° 

2245 

7320 

6.48 

2680 

44 

17 

9.539907 

5-69 

2198 

7709 

6.47 

2291 

43 

18 

9.540249 

5-69 

2151 

8098 

6.47 

1902 

42 

19 

0590 

5.68 

2105 

8486 

6.46 

1514 

41 

20 

0931 

5.68 

2058 

8873 

6.46 

1127 

40 

21 

22 

9.541272 
1613 

5-67 

9.972011 
1964 

9261 
9.569648 

6-45 
6.45 

0739 
10.430352 

39 
38 

23 

1953 

5^66 

1917 

9-570035 

6.45 

10.429965 

37 

24 

2293 

5.66 

1870 

0422 

6.44 

9578 

36 

25 

2632 

5.65 

1823 

0809 

6.44 

9191 

35 

26 

2971 

5.65 

1776 

•78 

1195 

6.43 

8805 

34 

27 

3310 

1729 

•79 

1581 

6.43 

8419 

33 

28 

3649 

5-4 

1682 

1967 

6.42 

8033 

32 

29 

3987 

1635 

2352 

6.42 

7648 

31 

30 

43*5 

5-63 

1588 

2738 

6.42 

7262 

30 

31 

9.544663 

5.62 

9.971540 

9.573123 

6.41 

10.426877 

29 

32 

5000 

5.62 

H93 

35°7 

6.41 

6493 

28 

33 
34 

5338 

5674 

5.61 
5.61 

1446 
1398 

389* 
4*76 

6.40 
6.40 

6108 
57*4 

27 
26 

35 

6011 

5.60 

I35i 

4660 

6-39 

534° 

25 

36 

6347 

5.60 

1303 

5°44 

6-39 

4956 

24 

37 

6683 

5-59 

1256 

54*7 

6-39 

4573 

23 

38 
39 
40 

7010 

7354 
7689 

5-59 
5-5* 

1208 
1161 
1113 

•79 

5810 
6193 
6576 

6.38 
6.38 
6.37 

4190 
3807 
34*4 

22 
21 
20 

^41 

9.548024 

5-57 

9.971066 

.80 

9.576958 

6-37 

10.423042 

19 

42 

8359 

5-57 

1018 

7341 

6.36 

2659 

18 

43 

8693 

5.56 

0970 

77*3 

6.36 

2277 

17 

44 

9027 

5-56 

0922 

8104 

6.36 

1896 

16 

45 

9360 

5-55 

0874 

8486 

6-35 

I5H 

15 

46 

9-549693 

5-55 

0827 

8867 

6-35 

"33 

14 

47 

9.550026 

5-54 

0779 

9248 

6.34 

0752 

13 

48 

°359 

5-54 

0731 

9-5796*9 

6-34 

10.420371 

12 

49 

0692 

5-53 

0683 

9.580009 

6-34 

10.419991 

11 

50 

102^ 

5-53 

0635 

0389 

6-33 

9611 

10 

61 

9-55*356 

5-5* 

9.970586 

9.580769 

6-33 

10.419231 

9 

52 

1687 

5.52 

0538 

1149 

6.32 

8851 

8 

53 

2018 

5-5* 

0490 

15*8 

6.32 

8472 

7 

54 

*349 

0442 

1907 

6.32 

8093 

6 

55 

2680 

5-5i 

0394 

.80 

2286 

6.31 

7714 

5 

56 

3010 

5-5° 

°345 

.81 

2665 

6.31 

7335 

4 

57 

58 

334i 
3670 

5-50 
5-49 

0297 
0249 

3°43 
34** 

6.30 
6.30 

6957 
6578 

3 
2 

59 

4000 

5-49 

O2OO 

.81 

3800 

6.29 

6200 

1 

60 

9-5543*9 

9.970152 

9-584I77 

10.415823 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l/ 

Cotang. 

Diff.l" 

Tang. 

M. 

110°                                      69° 

62 


21°       SINES  AND  TANOENTS.      158° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

0 

9-554329 

5.48 

9.970152 

.81 

9.584177 

6.29 

10.415823 

60 

1 
2 

4658 
4987 

5-47 

0103 
0055 

4555 
4932 

6.29 
6.28 

544J 
5068 

59 
58 

3 

5315 

5-47 

9.970006 

5309 

6.28 

4691 

57 

4 

5643 

5.46 

9.969957 

5686 

6.27 

43H 

56 

5 

597i 

5.46 

9909 

6062 

6.27 

3938 

55 

6 

6299 

5-45 

9860 

6439 

6.27 

3561 

54 

7 

6626 

5-45 

9811 

6815 

6.26 

3l85 

53 

8 

6953 

5-44 

9762 

7190 

6.26 

2810 

52 

9 

10 

7280 
7606 

5-44 
5-43 

9665 

.81 

7566 
7941 

6.25 
6.25 

M34 
2059 

51 
50 

11 

9-55793J 

5-43 

9.969616 

.82 

9-588316 

6.25 

10.411684 

49 

12 

8258 

5-43 

9567 

8691 

6.24 

1309 

48 

13 

8583 

5-4* 

9518 

9066 

6.24 

0934 

47 

14 

8909 

5-42 

9469 

944° 

6.23 

0560 

46 

15 

9*34 

5-41 

9420 

9.589814 

6.23 

10.410186 

45 

16 

9558 

5.41 

9370 

9.590188 

6.23 

10.409812 

44 

17 

9.559883 

5-4° 

9321 

0562 

6.22 

9438 

43 

18 

9.560207 

5-40 

9272 

°935 

6.22 

9065 

42 

19 
20 

0531 
0855 

5-39 
5-39 

9223 
9173 

1308 
1681 

6.22 
6.21 

8692 
8319 

41 
40 

21 

9.561178 

5.38 

9.969124 

9.592054 

6.21 

10.407946 

39 

22 

1501 

5-38 

9075 

2426 

6.  20 

7574 

38 

23 

1824 

5-37 

9025 

2798 

6.20 

7202 

37 

24 

2146 

5-37 

8976 

.82 

6.20 

6829 

36 

25 

2468 

5.36 

8926 

.83 

3542 

6.19 

6458 

35 

26 

2790 

S-36 

8877 

39*4 

6.19 

6086 

34 

27 

3112 

5.36 

8827 

4285 

6.18 

5715 

33 

28 

3433 

5-35 

8777 

4656 

6.18 

5344 

32 

29 

3755 

5-35 

8728 

5027 

6.18 

4973 

31 

30 

4°75 

5-34 

8678 

5398 

6.17 

4602 

30 

31 

32 

9.564396 
4716 

5-34 
5-33 

9.968628 

8578 

9.595768 
6138 

6.17 

6.16 

10.404232 
3862 

29 
28 

33 

5036 

5-33 

8528 

6508 

6.16 

3492 

27 

34 

5356 

5-3* 

8479 

6878 

6.16 

3122 

26 

35 

5676 

5-32 

8429 

7247 

6.15 

2753 

25 

36 

5995 

5.31 

8379 

7616 

6.15 

2384 

24 

37 

6314 

5.31 

8329 

7985 

6.15 

2015 

23 

38 

6632 

5-31 

8278 

•83 

8354 

6.14 

1646 

22 

39 

6951 

5-30 

8228 

.84 

8722 

6.14 

1278 

21 

40 

7269 

5-3° 

8178 

9091 

6.13 

0909 

20 

41 

9-567587 

5.29 

9.968128 

9459 

6.13 

0541 

19 

42 
43 

7904 
8222 

5.29 

5.28 

8078 
8027 

9.599827 
9.600194 

6.13 

6.12 

10.400173 
10.399806 

18 
17 

44 

8539 

5.28 

7977 

0562 

6.12 

9438 

16 

45 

8856 

5.28 

7927 

0929 

6.ii 

9071 

15 

46 

9172 

5-27 

7876 

1296 

6.ii 

8704 

14 

47 
48 

9488 
9.569804 

5.27 
5.26 

7826 
7775 

1662 
2029 

6.1  1 
6.10 

8338 
7971 

13 
12 

49 

9.570120 

5-26 

7725 

2395 

6.10 

7605 

11 

50 

0435 

7674 

2761 

6.10 

7^39 

10 

51 

9-57075I 

5-25 

9.967624 

9.603127 

6.09 

10.396873 

9 

52 

1066 

5.24 

7573 

.84 

3493 

6.09 

6507 

8 

53 

1380 

7522 

.85 

3858 

6.09 

6142 

7 

54 

1695 

5.23 

4223 

6.08 

5777 

6 

55 

2009 

7421 

4588 

6.08 

5412 

5 

56 

2323 

5-23 

737° 

4953 

6.07 

5°47 

4 

57 

2636 

5.22 

73*9 

53*7 

6.07 

4683 

3 

58 

2950 

5.22 

7268 

5682 

6.07 

4318 

2 

59 

3263 

5.21 

7217 

.85 

6046 

6.06 

3954 

1 

60 

9-573575 

9.967166 

9.606410 

10.393590 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

111°                                       68° 

63 


22°            LOGA 

157° 

RXTHMIC 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

I 

~0 

9-573575 

5.21 

9.967166 

.85 

9.606410 

6.06 

I°-39359° 

60 

1 

3888 

5.20 

7115 

6773 

6.06 

3227 

59 

2 

4200 

5.20 

7064 

7137 

6.05 

2863 

58  ! 

3 
4 
5 

4512 
4824 
S1!^ 

5-19 
5-19 
5.19 

7013 
6961 
6910 

7500 
7863 
8225 

6.05 
6.04 
6.04 

2500 
2137 
1775 

57 
56 
55 

6 

5447 

S.l8 

6859 

8588 

6.04 

1412 

54 

7 

5758 

5.18 

6808 

.85 

8950 

6.03 

1050 

53 

8 
9 
10 

6069 
6379 
6689 

5-17 
5-17 
5.16 

6756 
6705 
6653 

.86 

9312 
9.609674 
9.610036 

6.03 
6.03 
6.  02 

0688 
10.390326 
10.389964 

52 
51 
50 

11 

9.576999 

5.l6 

9.966602 

0397 

6.  02 

10.389603 

49 

12 

73°9 

5.16 

6550 

0759 

6.02 

9241 

48 

13 

7618 

6499 

1120 

6.01 

8880 

47 

14 

7927 

5-I5 

6447 

1480 

6.01 

8520 

46 

15 

8236 

5.14 

6395 

1841 

6.01 

8159 

45 

16 

8545 

5-H 

6344 

2201 

6.00 

7799 

44 

17 

8853 

6292 

2561 

6.00 

7439 

43 

18 

9162 

5-I3 

6240 

2921 

6.00 

7079 

42 

19 

9470 

5-J3 

6188 

3281 

5-99 

6719 

41 

20 

9-579777 

5.12 

6136 

.86 

3641 

5-99 

6359 

40 

21 

9.580085 

5.12 

9.966085 

.87 

9.614000 

5.98 

10.386000 

39 

22 

0392 

5-" 

6033 

4359 

5-98 

5641 

38 

23 

0699 

5.11 

4718 

5-98 

5282 

37 

24 

1005 

5928 

5077 

5-97 

4923 

36 

25 

1312 

5-io 

5876 

5435 

5-97 

4565 

35 

26 

1618 

5.10 

5824 

5793 

5-97 

4207 

34 

27 

1924 

5.09 

5772 

6151 

5.96 

3849 

33 

28 

2229 

5.09 

5720 

6509 

5.96 

349  * 

32 

29 

2535 

5-°9 

5668 

6867 

5.96 

31 

30 

2840 

5.08 

5615 

7224 

5-95 

2776 

30 

31 

9-583145 

5.08 

9.965563 

9.617582 

5-95 

10.382418 

29 

32 

3449 

5.07 

55" 

7939 

5-95 

2061 

28 

33 

3754 

5-°7 

5458 

8295 

5-94 

1705 

27 

34 

4058 

5.06 

5406 

.87 

8652 

5-94 

1348 

26 

35 

4361 

5-06 

5353 

.88 

9008 

5-94 

0992 

25 

36 

37 

4665 
4968 

S-o6 
5.05 

5248 

e  9364 
9.619721 

5-93 
5-93 

0636 
10.380279 

24 
23 

38 

5272 

5.05 

9.620076 

5-93 

10.379924 

22 

39 

5574 

5-°4 

5*43 

0432 

5-9* 

9568 

21 

40 

5877 

5.04 

5090 

0787 

5-92 

9213 

20 

41 

9.586179 

5-°3 

9.965037 

9.621142 

5.92 

10.378858 

19 

42 

6482 

5-°3 

4984 

H97 

5-91 

8503 

18 

43 

6783 

5-°3 

1852 

5.91 

8148 

17 

44 
45 

708) 
7386 

5.02 
5.02 

4879 
4826 

2207 
2561 

5-90 
5-9° 

7793 
7439 

16 
15 

46 

7688 

5.01 

4773 

2915 

5-9° 

7085 

14 

47 

7989 

5.01 

4719 

.88 

•1269 

5-89 

6731 

13 

48 

8289 

5.01 

4666 

.89 

3623 

5-89 

6377 

12 

49 

8590 

5.00 

4611 

3976 

5.89 

6024 

11 

50 

8890 

5.00 

4560 

433° 

5.88 

5670 

10 

51 

9.589190 

4.99 

9.964507 

9.624683 

5.88 

10.375317 

9 

52 

9489 

4-99 

•4454 

5036 

5-88 

4964 

8 

53 

9.589789 

4-99 

4400 

5388 

5-87 

4612 

7 

54 

9.590088 

4.98 

4347 

5.87 

4^59 

6 

55 

0387 

4.98 

4294 

6093 

5.87 

3907 

5 

56 

0686 

4-97 

4240 

6445 

5.86 

3555 

4 

57 

0984 

4-97 

4187 

6797 

5.86 

3203 

3 

58 

1282 

4-97 

4133 

7149 

5.86 

2851 

2 

59 

1580 

4.96 

4080 

.89 

7501 

5-85 

2499 

1 

60 

9.591878 

9.964026 

9.627852 

10.372148 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.1' 

Cotang. 

Diff.  1" 

Tang. 

M. 

112°                                       67° 

64. 


23°       SINES  AND  TANGENTS.      156° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.591878 

4.96 

9.964026 

.89 

9.627852 

5.85 

10.372148 

60 

1 

2176 

4-95 

397* 

.89 

8203 

1797 

59 

2 

2473 

4-95 

3919 

.89 

8554 

5-85 

1446 

58 

3 

2770 

4-95 

3865 

.90 

8905 

5-84 

1095 

57 

4 

3067 

4-94 

3811 

9*55 

5.84 

0745 

56 

5 

3363 

4-94 

3757 

9606 

5-83 

0394 

55 

6 

7 

3659 
3955 

4-93 
4-93 

37°4 
3650 

9.629956 
9.630306 

5.83 

10.370044 
10.369694 

54 
53 

8 

4251 

4-93 

3596 

0656 

5-83 

9344 

52 

9 

4547 

4.92 

3542 

1005 

5.82 

8995 

51 

10 

4842 

4.92 

3488 

1355 

5.82 

8645 

50 

11 

9-595*37 

4.91 

9-963434 

9.631704 

5-82 

10.368296 

49 

12 

5432 

4.91 

3379 

2053 

5.8l 

7947 

48 

13 

57*7 

4.91 

3325 

2401 

5.8l 

7599 

47 

14 

6021 

4.90 

327! 

2750 

5.8l 

7250 

46 

15 

63*5 

4.90 

3217 

3098 

5.80 

6902 

45 

16 

6609 

4-89 

3161 

.90 

3447 

5.8o 

6553 

44 

17 

6903 

4.89 

3108 

.91 

3795 

5.80 

6205 

43 

18 

7196 

4.89 

3°54 

4H3 

5-79 

5857 

42 

19 

749° 

4.88 

2999 

449° 

5-79 

5510 

41 

20 

7783 

4.88 

2945 

4838 

5-79 

5162 

40 

n 

9.598075 

4.87 

9.962890 

9.635185 

5.78 

10.364815 

39 

22 

8368 

4.87 

2836 

553* 

5.78 

4468 

38 

23 
24 

8660 
8952 

4.87 
4.86 

2781 
2727 

till 

5-78 
5-77 

4121 

3774 

37 
36 

25 

9244 

4.86 

2672 

6572 

5-77 

3428 

35 

26 

9536 

4-85 

2617 

6919 

5-77 

3081 

34 

27 

9.599827 

4.85 

2562 

7265 

5-77 

2735 

33 

28 

9.600118 

4.85 

2508 

7611 

5.76 

2389 

32 

29 
30 

0409 

0700 

4.84 
4.84 

H53 

2398 

.91 
.92 

7956 
8302 

5.76 
5.76 

2044 
1698 

31 
30 

31 

9.600990 

4.84 

9.962343 

9.638647 

5-75 

10.361353 

29 

32 

1280 

4.83 

2288 

8992 

5-75 

1008 

28 

33 

1570 

4.83 

2233 

9337 

5-75 

0663 

27 

34 

1860 

4.82 

2178 

9.639682 

5-74 

10.360318 

26 

35 

2150 

4.82 

2121 

9.640027 

5-74 

iQ-359973 

25 

36 

2439 

4.82 

2067 

0371 

5-74 

9629 

24 

37 

2728 

4.81 

2012 

0716 

5-73 

9284 

23 

38 

3017 

4.81 

1957 

1060 

5-73 

8940 

22 

39 

33°5 

4.81 

1902 

i4°-i 

5-73 

8596 

21 

40 

3594 

4.80 

1846 

1747 

5.72 

8253 

20 

41 

9.603882 

4.80 

9.961791 

9.642091 

5.72 

10.357909 

19 

42 

4170 

4-79 

1735 

2434 

5.72 

7566 

18 

43 

4457 

4-79 

l68o 

.92 

2777 

5-7* 

7223 

17 

44 

4745 

4-79 

1624 

•93 

3120 

5-71 

6880 

16 

45 

^5032 

4.78 

I56c 

3463 

5-71 

6537 

15 

46 

53*9 

4-78 

1511 

3806 

5-71 

6194 

14 

47 

5606 

4.78 

1458 

4148 

5-70 

5852 

13 

48 

5892 

4-77 

1402 

4490 

5.70 

5510 

12 

49 

6179 

4-77 

I346 

4832 

5-7° 

5168 

11 

50 

6465 

4.76 

1290 

5174 

5.69 

4826 

10 

51 

9.606751 

4.76 

9.961235 

9.645516 

5.69 

10.354484 

9 

52 

7036 

4.76 

"79 

5857 

5-69 

4'43 

8 

53 

7322 

4-75 

1123 

6199 

5-69 

3801 

7 

54 

7607 

4-75 

1067 

6540 

5.68 

3460 

6 

55 

7892 

4-74 

IOII 

6881 

5.68 

3"9 

5 

56 

8177 

4-74 

0955 

7222 

5.68 

2778 

4 

57 

8461 

4-74 

0899 

•93 

7562 

5.67 

2438 

3' 

58 

8745 

4-73 

0843 

•94 

7903 

5.67 

2097 

2 

59 

9029 

4-73 

0786 

•94 

8243 

5.67 

1757 

1 

60 

9.60931-; 

9.960730 

9.648581 

10.351417 

0 

Cosine. 

Diff.  1" 

Sine. 

Dial' 

Cotang. 

Diff.  V 

Tang. 

M. 

113°                                       66° 

65 


24°            IiOGA 

155° 

.RITHIMCIC 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.609313 

4-73 

9.960730 

•94 

9.648583 

5.66 

10.351417 

60 

1 

9597 

4.72 

0674 

8923 

5.66 

1077 

59 

2 

9.609880 

4.72 

0618 

9263 

5.66 

0737 

58 

3 

9.610164 

4.72 

0561 

9602 

5.66 

0398 

57 

4 

0447 

4.71 

0505 

9.649942 

5.65 

10.350058 

56  ' 

5 

0729 

4.71 

0448 

9.650281 

5.65 

10.349719 

55 

6 

JOI2 

4-7° 

0392 

0620 

5.65 

9380 

54 

7 

1294 

4.70 

°335 

°959 

5.64 

9041 

53 

8 

I576 

4.70 

0279 

1297 

5.64 

8703 

52 

9 

1858 

4.69 

0222 

1636 

5-64 

8364 

51 

10 

2140 

4.69 

0165 

•94 

1974 

5.63 

8026 

50 

11 

9.612421 

4.69 

0109 

•95 

9.652312 

5.63 

10.347688 

49 

12 

2702 

4.68 

9.960052 

2650 

735° 

48 

13 

2983 

4.68 

9-959995 

2988 

5-63 

7012 

47 

14 

3264 

4.67 

9938 

3326 

5-6* 

6674 

46 

15 

3545 

4.67 

9882 

3663 

5.62 

6337 

45 

16 

38*5 

4.67 

9825 

4000 

5.62 

6000 

44 

17 
18 
19 
20 

4105 

4385 
4665 

4944 

4.66 
4.66 
4.66 
4.65 

9768 

97" 
9654 
9596 

4337 
4674 
5011 
5348 

5.61 
5.61 
5.61 
5.61 

5663 
5326 

4989 
4652 

43 
42 
41 
40 

21 

9.615223 

4.65 

9-959539 

9.655684 

5.60 

10.344316 

39 

22 

55°2 

4.65 

9482 

6020 

5.60 

3980 

38 

23 

5781 

4.64 

9425 

6356 

5.60 

3644 

37 

24 

6060 

4.64 

9368 

•95 

6692 

5-59 

3308 

36 

25 

6338 

4.64 

9310 

.96 

7028 

5-59 

2972 

35 

26 

6616 

4.63 

9*53 

7364 

5-59 

2636 

34 

27 

6894 

4.63 

9'95 

7699 

5-59 

2301 

33 

28 

7172 

4.62 

9138 

8034 

5.58 

1966 

32 

29 

745° 

4.62 

9081 

8369 

5-58 

1631 

31 

30 

7727 

4.62 

9023 

8704 

5.58 

1296 

30 

31 

9.618004 

4.61 

9.958965 

9.659039 

5.58 

10.340961 

29 

32 

8281 

4.61 

8908 

9373 

5-57 

0627 

28 

33 
34 

8558 
8834 

4.61 
4.60 

8850 
8792 

9.659708 
9.660042 

5-57 
5-57 

10.340292 
10.339958 

27 
26 

35 

9110 

4.60 

8734 

0376 

5-57 

9624 

25 

36 

9386 

4.60 

8677 

0710 

5-56 

9290 

24 

37 

9662 

4-59 

8619 

1043 

8957 

23 

38 
39 

9.619938 
9.620213 

4-59 
4-59 

8561 
8503 

.96 
•97 

1377 
1710 

5-56 
5-55 

8623 
8290 

22 
21 

40 

0488 

4.58 

8445 

2043 

5-55 

7957 

20 

41 

0763 

4-58 

9-958387 

9.662376 

5-55 

10.337624 

19 

42 

1038 

4-57 

8329 

2709 

5-54 

7291 

18 

43 

1313 

4-57 

8271 

3042 

5-54 

6958 

17 

44 

1587 

4-57 

8213 

3375 

5-54 

6625 

16 

45 

1861 

4.56 

8154 

37°7 

5-54 

6293 

15 

46 

2135 

4-56 

8096 

4°39 

5-53 

5961 

14 

47 

2409 

4.56 

8038 

437i 

5-53 

5629 

13 

48 

2682 

4-55 

7979 

47°3 

5-53 

5297 

12 

49 

2956 

4-55 

7921 

5°35 

5-53 

4965 

11 

50 

3229 

4-55 

7863 

5366 

5-5* 

4634 

10 

51 

52 

9.623502 

3774 

4-54 
4-54 

9.957804 

7746 

•97 
.98 

9.665697 
6029 

5.52 
5-52 

10.334303 
3971 

8 

53 

4047 

4-54 

7687 

6360 

5-51 

3640 

7 

54 

4319 

4-53 

7628 

6691 

33°9 

6 

55 

4591 

4-53 

7570 

7021 

5.51 

2979 

5 

56 

4863 

4-53 

7511 

7352 

5-51 

2648 

4 

•57 

5135 

4.52 

745* 

7682 

5-5° 

2318 

3 

58 

5406 

4.52 

7393 

8013 

5-5° 

1987 

2 

59 

5677 

4-5* 

7335 

.98 

8343 

5-5° 

1657 

1 

60 

9.625948 

9.957276 

9.668672 

10.331328 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

114°                                       65° 

66 


25°       SINES  AND  TANGENTS.       154° 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.625948 

4.51 

9.957276 

.98 

9.668673 

5-5° 

10.331327 

60 

1 

6219 

4.51 

7217 

9002 

5-49 

0998 

59 

2 

6490 

4.51 

7158 

933* 

5-49 

0668 

58 

3 

6760 

4-50 

7099 

9661 

5-49 

0339 

57 

4 

7030 

4-5° 

7040 

9.669991 

5.48 

10.330009 

56 

5 

7300 

4-5° 

6981 

.98 

9.670320 

5.48 

10.329680 

55 

6 

7 

7570 
7840 

4.49 
4-49 

6921 
6862 

•99 

0649 

0977 

5-48 
5-48 

9351 
9023 

54 
53 

8 

8109 

4.49 

6803 

1306 

5-47 

8694 

52 

9 

8378 

4.48 

6  744 

1634 

5-47 

8366 

51 

10 

8647 

4.48 

6684 

1963 

5-47 

8037 

50 

11 

9.628916 

4-47 

9.956625 

9.672291 

5-47 

10.327709 

49 

12 

9185 

4-47 

6566 

2619 

5.46 

48 

13 

9453 

4-47 

6506 

2947 

7°53 

47 

14 

9721 

4.46 

6447 

3274 

5-46 

6726 

46 

15 

9.629989 

4.46 

6387 

3602 

5.46 

6398 

45 

16 

9.630257 

4.46 

6327 

39*9 

5-45 

6071 

44 

17 

0524 

4.46 

6268 

•99 

4*57 

5-45 

5743 

43 

18 

0792 

4-45 

6208 

1.  00 

4584 

5-45 

42 

19 

1059 

4-45 

6148 

4910 

5-44 

5090 

41 

20 

1326 

4-45 

6089 

5*37 

5-44 

4763 

40 

~2l 

9.631593 

4-44 

9.956029 

9.675564 

5-44 

10.324436 

39 

22 

1859 

4.44 

5969 

5890 

5-44 

4110 

38 

23 

2125 

4-44 

5909 

6216 

5-43 

3784 

37 

24 

2392 

4-43 

5849 

6543 

5-43 

3457 

36 

25 

2658 

4-43 

5789 

6869 

5-43 

S'31 

35 

26 

2923 

4-43 

57*9 

7*94 

5-43 

2806 

34 

27 

3189 

4.42 

5669 

7520 

2480 

33 

28 

3454 

4.42 

5609 

7846 

5.42 

2154 

32 

29 

3719 

4.42 

5548 

8171 

1829 

31 

30 

3984 

4.41 

5488 

1.  00 

8496 

5.42 

1504 

30 

~3T 

9.634249 

4.41 

9.955428 

I.OI 

9.678821 

5.41 

10.321179 

29 

32 

45H 

4.40 

5368 

9146 

5-41 

0854 

28 

33 

4778 

4.40 

53°7 

f  947i 

5.41 

0529 

27 

34 
35 

5042 
5306 

4.40 
4-39 

5*47 

5186 

9.679795 
9.680120 

5.41 

5-4° 

10.320205 
10.319880 

26 
25 

36 

557° 

4-39 

5126 

0444 

5-4° 

9556 

24 

37 

5834 

4-39 

5°65 

0768 

5-4° 

9232 

23 

38 

6097 

4-39 

5°°5 

1092 

5-4° 

8908 

22 

39 

6360 

4-38 

4944 

1416 

5-39 

8584 

21 

40 

6623 

4.38 

4883 

1740 

5-39 

8260 

20 

41 

9.636886 

4-37 

9.954823 

9.682063 

5-39 

10.317937 

19 

42 

7148 

4-37 

4762 

2387 

5-39 

7613 

18 

43 

7411 

4-37 

4701 

2710 

7290 

17 

44 

7673 

4-37 

4640 

3°33 

"I 

6967 

16 

45 

7935 

4.36 

4579 

'  I.OI 

3356 

6644 

15 

46 

8197 

4-36 

4518 

1.  02 

3679 

5-38 

6321 

14 

47 

8458 

4-36 

4457 

4001 

5-37 

5999 

13 

48 

8720 

4-35 

4396 

4324 

5-37 

5676 

12 

49 

8981 

4-35 

4335 

4646 

5-37 

5354 

11 

50 

9242 

4-35 

4274 

4968 

5-37 

10 

51 

52 

9503 
9.639764 

4-34 
4-34 

9.954213 

9.685290 
5612 

5.36 

10.314710 
4388 

9 

8 

53 

9.640024 

4-34 

4090 

5934 

5-36 

4066 

7 

54 

0284 

4-33 

4029 

6255 

5.36 

3745 

6 

55 

0544 

4-33 

3968 

6577 

5-35 

3423 

5 

56 

0804 

4-33 

3906 

6898 

5-35 

3102 

4 

57 

1064 

4-3* 

3845 

7219 

5-35 

2781 

3 

58 

1324 

3783 

1.02 

7540 

5-35 

2460 

2 

59 
60 

1583 
9.641842 

4-32 

3722 
9.953660 

1.03 

7861 
9.688182 

5-34 

2139 
10.311818 

1 
0 

Cosine. 

Diff.  l" 

Sine. 

Diff.l"   Cotang. 

Diff.  I" 

Tang. 

M. 

115°                                       64° 

26°            LOGARITHMIC           153° 

M. 

Sine. 

Difif.  I" 

Cosine. 

Difif.  1" 

Tang. 

Difif.  1" 

Cotang. 

0 

9.641842 

4-31 

9.953660 

1.03 

9.688182 

5-34 

10.311818 

60 

1 

2101 

4.31 

3599 

8502 

5-34 

I498 

59  , 

2 

3 

2360 
26l8 

4.31 
4-3° 

3537 
3475 

8823 
9H3 

5-34 
5-33 

1177 
0857 

58 
57 

4 

2877 

4.30 

3413 

9463 

5-33 

0537 

56 

5 

4-3° 

335* 

9.689783 

5-33 

10.310217 

55 

6 

3393 

4.30 

3290 

9.690103 

5-33 

10.309897 

54 

7 

365° 

4.29 

3228 

0423 

5-33 

9577 

53 

8 

3908 

4.29 

3166 

0742 

5-32 

9258 

52 

9 

4165 

4.29 

3104 

1062 

5-3* 

8938 

51 

10 

4423 

4.28 

3042 

1-03 

1381 

5-3* 

8619 

50 

11 
12 

9.644680 
4936 

4.28 
4.28 

9.952980 
2918 

1.04 

9.691700 
2019 

5-31 
5-31 

10.308300 
7981 

49 

48 

13 

4.27 

2855 

2338 

7662 

47 

14 

5450 

4.27 

2793 

2656 

5-31 

7344 

46 

15 

5706 

4.27 

2731 

2975 

7025 

45 

16 
17 

5962 
6218 

4.26 
4.26 

2669 
2606 

3*93 
3612 

5-3° 
5-3° 

6707 
6388 

44 

43 

18 

6474 

4.26 

2544 

393° 

5-3° 

6070 

42 

19 

6729 

4.26 

2481 

4248 

5-3° 

5752 

41 

20 

6984 

4.25 

2419 

4566 

5434 

40 

21 

9.647240 

4.25 

9.952356 

9.694883 

5.29 

10.305117 

39 

22 

7494 

4.24 

2294 

5201 

5.29 

4799 

38 

23 

7749 

4.24 

2231 

1.04 

5518 

5>29 

4482 

37 

24 

8004 

4.24 

2168 

1.05 

5836 

5-29 

4164 

36 

25 

8258 

4.24 

2106 

5.28 

3847 

35 

26 

8512 

4.23 

2043 

6470 

5.28 

353° 

34 

27 

8766 

4.23 

1980 

6787 

5.28 

3213 

33 

28 

9020 

4-23 

1917 

7103 

5.28 

2897 

32 

29 

9274 

4.22 

.  1854 

7420 

2580 

31 

30 

9527 

4.22 

1791 

7736 

5-*7 

2264 

30 

31 
32 

9.649781 
9.650034 

4.22 
4.22 

9.951728 
1665 

9.698053 
8369 

5-*7 

10.301947 
1631 

29 

28 

33 

0287 

4.21 

1602 

8685 

5^6 

1315 

27 

34 

°539 

4.21 

1539 

9001 

5-26 

0999 

26 

35 

0792 

4.21 

1476 

9316 

5.26 

0684 

25 

36 

1044 

4.20 

1412 

1.05 

9632 

5.26 

0368 

24 

37 

1297 

4.20 

1349 

1.  06 

9.699947 

5.26 

10.300053 

23 

38 

1549 

4.20 

1286 

9.700263 

5-25 

10.299737 

22 

39 

1800 

4.19 

1222 

0578 

5.25 

9422 

21 

40 

2052 

4.19 

1159 

0893 

5-25 

9107 

20 

41 

9.652304 

4.19 

9.951096 

9.701208 

5.24 

10.298792 

19 

42 

2555 

4.18 

1032 

15*3 

5-^4 

8477 

18 

43 

2806 

4.18 

0968 

1837 

5.24 

8163 

17 

44 

3°57 

4.18 

0905 

2152 

7848 

16 

45 

33°8 

4.18 

0841 

2466 

5-^4 

7534 

15 

46 

3558 

4.17 

0778 

2780 

5-23 

7220 

14 

47 

3808 

4.17 

0714 

3°95 

5-23 

6905 

13 

48 

4059 

4.17 

0650 

3409 

5-23 

6591 

12 

49 

4309 

4.16 

0586 

1.  06 

37*3 

5-23 

6277 

11 

50 

4558 

4.16 

0522 

1.07 

4036 

5.22 

5964 

10 

51 

9.654808 

4.16 

9.950458 

9.704350 

5.22 

10.295650 

9 

52 

5058 

4.16 

0394 

4663 

5.22 

5337 

8 

53 

53°7 

4.15 

0330 

4977 

5.22 

5023 

7 

54 

5556 

4.15 

0266 

5290 

5.22 

4710 

6 

55 

5805 

4.15 

02O2 

5603 

5.21 

4397 

5 

56 

6054 

4.14 

0138 

5916 

5.21 

4084 

4 

57 

6302 

4.14 

0074 

6228 

5.21 

377* 

3 

58 

6551 

4.14 

9.950010 

6541 

5.21 

3459 

2 

59 

6799 

4.13 

9.949945 

I.07 

6854 

5.21 

3146 

1 

60 

9.657047 

9.949881 

9.707166 

10.292834 

0 

Cosine. 

Diff.  I" 

Sine. 

Difif.  V 

Cotang. 

Difif.  I" 

Tang. 

M. 

116°                                       63° 

68 


27°       SINES  AND  TANGENTS.      152° 

M. 

•Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

0 

9.657047 

4.13 

9.949881 

1.07 

9.707166 

5-20 

10.292834 

60 

1 

7295 

4-13 

9816 

1.07 

7478 

5.20 

2522 

59 

2 

7542 

4.12 

9752 

1.07 

7790 

5.20 

2210 

58 

3 

779° 

4.12 

9688 

1.  08 

8102 

5.20 

1898 

57 

:  4 

8037 

4.12 

9623 

8414 

1586 

56 

i  5 

8284 

4.12 

9558 

8726 

5.19 

1274 

55 

6 

853J 

4.11 

9494 

9°37 

5.19 

0963 

54 

7 

8778 

4.11 

9429 

9349 

5.19 

0651 

53 

8 

9025 

4.11 

9364 

9660 

5-19 

0340 

52 

9 

9271 

4.10 

9300 

9.709971 

5.18 

IO.290029 

51 

10 

9517 

4.10 

9235 

9.710282 

5.!8 

10.289718 

50 

11 

9.659763 

4.10 

9.949170 

°593 

5.18 

9407 

49 

B 

9.660009 

4.09 

9105 

0904 

5.18 

9096 

48 

13 

0255 

4.09 

9040 

1215 

5.18 

8785 

47 

14 

0501 

4.09 

8975 

1525 

8475 

46 

15 

0746 

4.09 

8910 

1836 

5-17 

8l64 

45 

16 

0991 

4.08 

8845 

1.08 

2146 

5-1? 

7854 

44 

17 

1236 

'  4.08 

8780 

1.09 

2456 

7544 

43 

18 

1481 

4.08 

8715 

2766 

5.16 

7^34 

42 

19 

1726 

4.07 

8650 

3076 

5.16 

6924 

41 

20 

1970 

4.07 

8584 

3386 

5.16 

6614 

40 

21 

9.662214 

4.07 

9.948519 

9.713696 

5.16 

10.286304 

39 

22 

2459 

4.07 

8454 

4005 

5-i6 

5995 

38 

23 

2703 

4-06 

8388 

43  '4 

5-15 

5686 

37 

24 

2946 

4.06 

8323 

4624 

5-15 

5376 

36 

25 

3190 

4.06 

8257 

4933 

5-J5 

5067 

35 

26 

3433 

4.05 

8192 

5242 

5.15 

4758 

34 

27 

3677 

4-°5 

8126 

5-H 

4449 

33 

28 

3920 

4.05 

8060 

1.09 

5860 

5.14 

4140 

32 

29 

4163 

4.05 

7995 

1.  10 

6168 

5-H 

3832 

31 

30 

4406 

4.04 

7929 

6477 

5.14 

35^3 

30 

31 

9.664648 

4.04 

9.947863 

9.716785 

5-H 

10.283215 

29 

32 

4891 

4.04 

7797 

7093 

2907 

28 

33 

5'33 

4.03 

7731 

7401 

5.13 

2599 

27 

34 

5375 

4.03 

7665 

7709 

5.13 

2291 

26 

35 

4.03 

7600 

8017 

5-x3 

1983 

25 

36 

5859 

4.02 

7533 

83^5 

5.13 

1675 

24 

37 

6100 

4.02 

7467 

8633 

5.12 

1367 

23 

38 

6342 

4.02 

7401 

8940 

5.12 

1060 

22 

39 

6583 

4.02 

7335 

9248 

5.12 

0752 

21 

40 

6824 

4.01 

7269 

9555 

5.12 

0445 

20 

41 

9.667065 

4.01 

9.947203 

1.  10 

9.719862 

5.12 

10.280138 

19 

42 

7305 

4.01 

7136 

i.  ii 

9.720169 

5.11 

10.279831 

18 

43 

7546 

4.01 

7070 

0476 

5-" 

9524 

17 

44 

7786 

4.00 

7004 

0783 

5.11 

9217 

16 

45 

8027 

4.00 

6937 

1089 

5-11 

8911 

15 

46 

8267 

4.00 

6871 

1396 

5.11 

8604 

14 

47 

8506 

3-99 

6804 

1702 

5.10 

8298 

13 

48 

8746 

3-99 

6738 

2009 

5.10 

7991 

12 

49 
50 

8986 
9225 

3-99 
3-99 

6671 
6604 

2315 
2621 

5.10 
5.10 

7685 
7379 

11 
10 

51 

9464 

3.98 

9.946538 

9.7229*7 

5.10 

10.277073 

9 

52 
53 

9703 
9.669942 

3-98 

6471 
6404 

3*3* 
3538 

5.09 
5.09 

6768 
6462 

8 

7 

54 

9.670181 

3-97 

6337 

i.  ii 

3844 

5.09 

6156 

6 

55 

0419 

3-97 

6270 

1.  12 

4149 

5.09 

5851 

5 

56 

0658 

3-97 

6203 

4454 

5.09 

5546 

4 

57 

0896 

3-97 

6136 

4759 

5.08 

5241 

3 

58 

"34 

3-96 

6069 

5°65 

5.08 

4935 

2 

59 

1372 

6002 

1.  12 

5369 

5.08 

463* 

1 

60 

9.671609 

9-945935 

9.725674 

10.274326 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang. 

Diff.  I" 

Tang. 

M.  ! 

117°                                     62° 

26 


69 


28°            LOGARITHMIC           151° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang.  - 

~~o 

9.671609 

3-96 

9-945935 

1.  12 

9.725674 

5-08 

10.274326 

60 

1 

1847 

3-95 

5868 

5979 

5.08 

4021 

59 

2 

2084 

3-95 

5800 

6284 

5-°7 

3716 

58 

3 

232J 

3-95 

5733 

6588 

5-°7 

3412 

57 

4 

2558 

3-95 

5666 

6892 

5-07 

3108 

56 

5 

2795 

3-94 

5598 

7197 

5.07 

2803 

55 

6 

3°2o 

3-94 

553i 

1.  12 

7501 

5-07 

2499 

54 

7 

3268 

3-94 

5464 

I.I3 

7805 

5.06 

2195 

53 

8 

3505 

3-94 

5396 

8109 

5.06 

1891 

52 

9 

3741 

3-93 

53^8 

8412 

5.06 

1588 

51 

10 

3977 

3-93 

5261 

8716 

5.06 

1284 

50 

11 

9.674213 

3-93 

9-945I93 

9.729020 

5.06 

10.270980 

49 

12 

4448 

3-9* 

51*5 

93*3 

5-°5 

0677 

48 

13 

4684 

3-9* 

5°58 

9626 

5-°5 

0374 

47 

14 

4919 

3-9* 

499° 

9.729929 

5-°5 

10.270071 

46 

15 

5J55 

3.92 

4922 

9.730233 

5-°5 

10.269767 

45 

16 

5390 

3-91 

4854 

°535 

5-°5 

9465 

44 

17 

5624 

3-91 

4786 

0838 

5-°4 

9162 

43 

18 

5859 

3-91 

4718 

1141 

5.04 

8859 

42 

19 

6094 

3-9i 

4650 

I.I3 

1444 

5.04 

8556 

41 

20 

6328 

3-9° 

4582 

I.I4 

1746 

5-°4 

8254 

40 

~21 

9.676562 

3.90 

9.944514 

9.732048 

5-°4 

10.267952 

39 

22 

6796 

3-9° 

4446 

235i 

5-°3 

7649 

38 

23 

7030 

3-9° 

4377 

2653 

5-°3 

7347 

37 

24 

7264 

3-89 

43°9 

2955 

5-°3 

7045 

36 

25 

749  8 

3-89 

4.241 

3*57 

5-03 

6743 

35 

26 

7731 

3-89 

4172 

3558 

5-03 

6442 

34 

27 

7964 

3.88 

4104 

3860 

5.02 

6140 

33 

28 

8197 

3.88 

4036 

4162 

5.02 

5838 

32 

29 

8430 

3.88 

3967 

4463 

5.02 

5537 

31 

30 

8663 

3.88 

3899 

4764 

5.02 

5236 

30 

31 

9.678895 

3-87 

9.943830 

9.735066 

5.02 

10.264934 

29 

32 

9128 

3.87 

3761 

I.I4 

5367 

5.02 

4633 

28 

33 

9360 

3.87 

3693 

I.I5 

5668 

5.01 

4332 

27 

34 

959^ 

3-87 

3624 

5969 

5.01 

4031 

26 

35 

9.679824 

3-86 

3555 

6269 

5.01 

3731 

25 

%36 

9.680056 

3.86 

3486 

6570 

5.01 

343° 

24 

37 

0288 

3.86 

3417 

6871 

5.01 

3129 

23 

38 

°5!9 

3-85 

3348 

7171 

5.00 

2829 

22 

39 

0750 

3.85 

3279 

747i 

5.00 

2529 

21 

40 

0982 

3-85 

3210 

7771 

5.00 

2229 

20 

41 

9.681213 

3.85 

9.943141 

9.738071 

5.00 

10.261929 

19 

42 

H43 

3.84 

3072 

8371 

5.00 

1629 

18 

43 

1674 

3.84 

3003 

8671 

4-99 

1329 

17 

44 

1905 

3.84 

2934 

8971 

4-99 

1029 

16 

45 

2135 

3-84 

2864 

I-I5 

9271 

4-99 

0729 

15 

46 

2365 

3-83 

2795 

1.16 

9570 

4-99 

0430 

14 

47 

2595 

3-83 

2726 

9.739870 

4-99 

10.260130 

13 

48 

2825 

3-83 

2656 

9.740169 

4-99 

10.259831 

12 

49 

3°55 

3-83 

2587 

0468 

4.98 

9532 

11 

50 

3284 

3.82 

2517 

0767 

4-98 

9233 

10 

51 

9.683514 

3.82 

9.942448 

9.741066 

4-98 

10.258934 

9 

52 

3743 

3.82 

2378 

i365 

4.98 

863*; 

8 

53 

3972 

3.82 

2308 

1664 

4-98 

8336 

7 

54 

4201 

3.81 

2239 

1962 

4-97 

8038 

6 

55 

443° 

3-8i 

2169 

2261 

4-97 

7739 

5 

56 

4658 

3.81 

2099 

2559 

4-97 

7441 

4 

57 

4887 

3.80 

2029 

2858 

4-97 

7142 

3 

58 

5U5 

3.80 

1959 

1.16 

3156 

4-97 

6844 

2 

59 
60 

f  5343 
9.685571 

3.80 

1889 
9.941819 

1.17 

3454 
9.743752 

4-97 

6546 
10.256248 

1 
0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

118°                                         61° 

70 


29°       SINES  AND  TANGENTS.       150° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  I" 

Cotang. 

0 

9.685571 

3.80 

9.941819 

1.17 

9-743752 

4.96 

10.256248 

60 

1 

5799 

3-79 

1749 

4050 

4.96 

5950 

59 

2 

6027 

3-79 

1679 

4348 

4.96 

5652 

58 

3 

6254 

3-79 

1609 

4645 

4.96 

5355 

57 

4 
5 

6482 
6709 

3-79 
3.78 

1539 
1469 

4943 
5240 

4.96 
4-95 

5°57 
4760 

56 
55 

6 

6936 

3-78 

1398 

5538 

4-95 

4462 

54 

7 

7163 

3.78 

1328 

5835 

4-95 

4165 

53 

8 

7389 

3-78 

"1258 

6132 

4-95 

3868 

52 

9 

7616 

3-77 

1187 

6429 

4-95 

3571 

51 

10 

7843 

3-77 

1117 

I.I7 

6726 

4-95 

3274 

50 

11 

9.688069 

3-77 

9.941046 

1.18 

9.747023 

4-94 

10.252977 

49 

12 

8295 

3-77 

°975 

7319 

4-94 

2681 

48 

13 

8521 

3.76 

0905 

7616 

4-94 

2384 

47 

14 

8747 

3-76 

0834 

79T3 

4-94 

2087 

46 

15 

8972 

3.76 

0763 

8209 

4-94 

1791 

45 

16 

9198 

3.76 

0693 

8505 

4-93 

*495 

44 

17 

9423 

3-75 

0622 

8801 

4-93 

1199  43 

18 

9648 

3-75 

0551 

9097 

4-93 

0903 

42 

19 

9.689873 

3-75 

0480 

9393 

4-93 

0607 

41 

20 

9.690098 

3-75 

0409 

9689 

4-93 

0311 

40 

21 

0323 

3-74 

9.940338 

9.749985 

4-93 

10.250015 

39 

22 

0548 

3-74 

0267 

9.750281 

4-93 

10.249719 

38 

23 

0772 

3-74 

0196 

1.18 

0576 

4.92 

9424 

37 

24 

0996 

3-74 

0125 

1.19 

0872 

4.92 

9128 

36 

25 

1220 

3-73 

9-940054 

1167 

4.92 

8833 

35 

26 

1444 

3-73 

9.939982 

1462 

4.92 

8538 

34 

27 

1668 

3-73 

9911 

1757 

4.92 

8243 

33 

28 

1892 

3-73 

9840 

2052 

4.91 

7948 

32 

29 

2115 

3-72 

9768 

2347 

4.91 

7653 

31 

30 

*339 

3-72 

9697 

2642 

4.91 

7358 

30 

~31 

9.692562 

3-72 

9.939625 

9-75*937 

4.91 

10.247063 

29 

32 

2785 

3.71 

9554 

3231 

4.91 

6769 

28 

33 

3008 

3-7i 

9482 

3526 

4.91 

6474 

27 

34 

3231 

3-7i 

9410 

3820 

4.90 

6180 

26 

35 

3453 

3-7i 

9339 

1.19 

4"5 

4.90 

5885 

25 

36 

3676 

3-7° 

9267 

1.20 

4409 

4.90 

5591 

24 

37 

3898 

3-70 

9'95 

4703 

4.90 

5297 

23 

38 

4120 

3-70 

9123 

4997 

4.90 

5003 

22 

39 

4342 

3-7° 

9052 

5291 

4-9° 

4709 

21 

40 

4564 

3.69 

8980 

5585 

4.89 

44i5 

20 

41 

9.694786 

3-69 

9.938908 

9-755878 

4.89 

10.244122 

19 

42 

5007 

3-69 

8836 

6172 

4.89 

3828 

18 

43 

5229 

3.69 

8763 

6465 

4.89 

3535 

17 

44 

545° 

3.68 

8691 

6759 

4.89 

3241 

16 

45 

5671 

3.68 

8619 

7052 

4.89 

2948 

15 

46 

5892 

3.68 

8547 

7345 

4.88 

2655 

14 

47 

6113 

3.68 

8475 

1.20 

7638 

4.88 

2362 

13 

48 

6334 

3-67 

8402 

I.2I 

7931 

4.88 

2069 

12 

49 

6554 

3.67 

8330 

8224 

4.88 

1776 

11 

50 

6775 

3.67 

8258 

8517 

4.88 

1483 

10 

51 

9.696995 

3.67 

9.938185 

9.758810 

4.88 

10.241190 

9 

52 

7215 

3-66 

8113 

9102 

4.87 

0898 

8 

53 

7435 

3.66 

8040 

9395 

4-87 

0605 

7 

54 

7654 

3.66 

7967 

9687 

4.87 

0313 

6 

55 

7874 

3.66 

7895 

9-759979 

4.87 

10.240021 

5 

56 

8094 

3.65 

7822 

9.760272 

4.87 

10.239728 

4 

57 

8313 

3-65 

7749 

0564 

4-87 

943  6 

3 

58 

853^ 

3-65 

7676 

0856 

4.86 

9144 

2 

59 

8751 

3-65 

7604 

I.2I 

1148 

4.86 

8852 

1 

60 

9.698970 

9.937531 

9.761439 

10.238561 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang.  |  Diff.  1" 

Taug. 

M. 

119°                                       60° 

71 


30°            LOaARITHMIC           149° 

M.  |   Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.698970 

3.64 

9-937531 

1.  21 

9.761439 

4-86 

10.238561 

60 

1 

9189 

3-64 

7458 

1.22 

1731 

4.86 

8269 

59 

2 

9407 

3.64 

7385 

2023 

4.86 

7977 

58 

3 
4 

9626 
9.699844 

3-64 
3.63 

7312 
7238 

2606 

4.86 

7686 
7394 

57 
56 

5 

9.700062 

3.63 

7165 

2897 

4-85 

7103 

55 

6 

0280 

3-63 

7092 

3188 

4.85 

6812 

54 

7 

0498 

3.63 

7019 

3479 

4-85 

6521 

53 

8 

0716 

3.63 

6946 

3770 

4-85 

6230 

52 

9 

°933 

3.62 

6872 

4061 

4.85 

5939 

51 

10 

"51 

3.62 

6799 

435* 

4.84 

5648 

50 

TT 

9.701368 

3.62 

9.936725 

1.22 

9.764643 

4.84 

10.235357 

49 

12 

1585 

3.62 

6652 

1.23 

4933 

4.84 

5067 

48 

13 

1802 

3.61 

6578 

5224 

4.84 

4776 

47 

14 

2019 

3.61 

6505 

55H 

4.84 

4486 

46 

15 

2*136 

3.61 

6431 

5805 

4-84 

4195 

45 

16 

2452 

3.61 

6357 

6095 

4.84 

39°5 

44 

17 

2669 

3.60 

6284 

6385 

4.83 

36l5 

43 

18 

2885 

3.60 

6210 

6675 

4.83 

33*5 

42 

19 

3101 

3.60 

6136 

6965 

4.83 

3°35 

41 

20 

33*7 

3.60 

6062 

7^55 

4-83 

2745 

40 

21 

9-703533 

3-59 

9.935988 

9.767545 

4-83 

10.232455 

39 

22 

3749 

3-59 

59H 

7834 

4.83 

2166 

38 

23 

3964 

3-59 

1.23 

8124 

4.82 

1876 

37 

24 

4J79 

3-59 

5766 

1.24 

8413 

4.82 

1587 

36 

25 

4395 

3-59 

5692 

8703 

4.82 

1297 

35 

28 

4610 

3-58 

5618 

8992 

4.82 

1008 

34 

27 
28 

4825 
5040 

3-58 

5543 
5469 

9281 
957° 

4.82 
4.82 

0719 
0430 

33 
32 

29 
30 

5*54 
5469 

3-58 
3-57 

5395 
5320 

9.769860 
9.770148 

4.81 
4.81 

10.230140 
10.229852 

31 
30 

31 

9.705683 

3-57 

9.935246 

°437 

4.81 

9563 

29 

32 

5898 

3-57 

5*71 

0726 

4.81 

9274 

28 

33 

6112 

3-57 

5°97 

1015 

4.81 

8985 

27 

34 

6326 

3-56 

5022 

1303 

4.81 

8697 

26 

35 

6539 

3.56 

4948 

1592 

4.81 

8408 

25 

36 

6753 

3  56 

4873 

1.24 

1880 

4.80 

8120 

24 

37 

6967 

3.56 

4798 

1.25 

2168 

4.80 

7832 

23 

38 

7180 

3-55 

4723 

2457 

4.80 

7543 

22 

39 

7393 

3-55 

4649 

2745 

4.80 

7255 

21 

40 

7606 

3-55 

4574 

3°33 

4.80 

6967 

20 

41 

9.707819 

3-55 

9-934499 

9.773321 

4.80 

10.226679 

19 

42 

8032 

3-54 

4424 

360? 

4-79 

6392 

18 

43 

8245 

3-54 

4349 

3896 

4-79 

6104 

17 

44 

8458 

3-54 

4274 

4184 

4-79 

5816 

16 

45 

8670 

3-54 

4199 

4-79 

5529 

15 

46 

8882 

3-53 

4121 

4759 

4-79 

5*41 

14 

47 

9094 

3-53 

4048 

5046 

4-79 

4954 

13 

48 

9306 

3-53 

3973 

1.25 

5333 

4-79 

4667 

12 

49 

9518 

3-53 

3898 

1.26 

5621 

4-78 

4379 

11 

50 

973° 

3-53 

3822 

5908 

4-78 

4092 

10 

51 

9.709941 

3-52 

9-933747 

9.776195 

4.78 

10.223805 

9 

52 

9.710153 

3-5* 

3671 

6482 

4.78 

35*8 

8 

53 

0364 

3-5* 

3596 

6769 

4.78 

3231 

7 

54 

°575 

3-52 

3520 

7°55 

4-78 

2945 

6 

55 

0786 

3-5i 

3445 

7342 

4-78 

2658 

5 

56 

0997 

3.51 

3369 

7628 

4-77 

2372 

4 

57 

1208 

3.51 

3*93 

79*5 

4-77 

2085 

3 

58 

14.19 

3-51 

3217 

8201 

4-77 

1799 

2 

59 

1629 

3-5° 

1.26 

8487 

4-77 

1513 

1 

60 

9.711839 

9.933066 

9.778774 

10.221226 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.]' 

Cotang. 

Diff.  1" 

Tang. 

M. 

120°                                       59° 

72 


31°       SINES  AND  TANGENTS.      148° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

0 

9.711839 

3-50 

9.933066 

1.26 

9.778774 

4-77 

10.221226 

60 

1 

2050 

3-50 

2990 

1.27 

9060 

4-77 

0940  59 

2 

2260 

3.50 

2914 

9346 

4.76 

0654  58 

3 

2469 

3-49 

2838 

9632 

4.76 

03681  57 

4 

2679 

3-49 

2762 

9.779918 

4.76 

10.220082!  56 

5 

2889 

3-49 

2685 

9.780203 

4.76 

10.2197971  55 

6 

3098 

3-49 

2609 

0489 

4-76 

9511  54 

7 

3308 

3-49 

2533 

0775 

4.76 

9225!  53 

8 

3517 

3-48 

2457 

1060 

8940  i  52 

9 

3726 

3-48 

2380 

1346 

4-75 

8654!  51 

10 

3935 

3-48 

2304 

1631 

4-75 

8369  50 

11 

9.714144 

3-48 

9.932228 

9-78I9I6 

4-75 

10.218084  49 

12 

435* 

3-47 

2151 

1.27 

22OI 

4-75 

7799  48 

13 

4561 

3-47 

2075 

1.28 

2486 

4-75 

75J4 

47 

14 

4769 

3-47 

I998 

2771 

4-75 

7229 

46 

15 

4978 

3-47 

1921 

4-75 

6944 

45 

16 

5186 

3-47 

1845 

3341 

4-75 

6659!  44 

17 

18 

5394 
5602 

3$ 

1768 
1691 

3626 
3910 

4-74 
4-74 

6374 
6090 

43 
42 

19 

5809 

3.46 

1614 

4'95 

4-74 

5805 

41 

20 

6017 

3.46 

X537 

4479 

4-74 

55*1 

40 

21 

9.716224 

3-45 

9.931460 

9.784764 

4-74 

10.215236 

39 

22 
23 

643* 
6639 

3-45 
3-45 

1383 
1306 

1.28 

5048 
533* 

4-74 
4-73 

495* 
4668 

38 
37 

24 

6846 

3-45 

1229 

J.29 

5616 

4-73 

4384 

36 

25 

7053 

3-45 

1152 

5900 

4-73 

4100 

35 

26 
27 

7*59 
7466 

3-44 
3-44 

1075 
0998 

6184 
6468 

4-73 
4-73 

3816 

353* 

34 
33 

28 
29 

7673 
7879 

3-44 
3-44 

0921 
0843 

6752 
7036 

4-73 
4-73 

3248!  32 
2964!  31 

30 

8085 

3-43 

0766 

4-7* 

2681  30 

31 

9.718291 

3-43 

9.930688 

9.787603 

4.72 

10.212397 

29 

32 

8497 

3-43 

0611 

7886 

4.72 

2114 

28 

33 

8703 

3-43 

°533 

8170 

4-7* 

1830 

27 

34 

8909 

3-43 

0456 

8453 

4-7* 

1547!  26 

35 

9114 

3-4* 

0378 

1.29 

8736 

4-7* 

1264  25 

36 

9320 

3.42 

0300 

1.30 

9019 

4.72 

0981  24 

37 

95*5 

3-4* 

0223 

9302 

4.71 

0698 

23 

38 

9730 

3.42 

0145 

9585 

4.71 

0415 

22 

39 

9-7I9935 

9.930067 

9.789868 

4.71 

10.2101321  21 

40 

9.720140 

3-41 

9.929989 

9.790151 

4.71 

10.209849 

20 

41 

0345 

"..S4* 

9911 

°433 

4.71 

9567 

19 

42 

0549 

3-41 

9833 

0716 

4.71 

9284 

18 

43 

0754 

3-40 

9755 

0999 

4.71 

9001 

17 

44 

0958 

3-4° 

9677 

1281 

4.71 

8719 

16 

45 

1162 

3-4° 

9599 

X563 

4.70 

8437 

15 

46 

1366 

3-4° 

95*i 

1846 

4.70 

8154 

14 

47 

1570 

3-40 

944* 

1.30 

2128 

4.70 

7872 

13 

48 

1774 

3-39 

9364 

2410 

4.70 

7590 

12 

49 

1978 

3-39 

9286 

2692 

4.70 

7308 

11 

50 

2181 

3-39 

9207 

2974 

4.70 

7026 

10 

51 

9.722385 

3-39 

9.929129 

9.793256 

4-7° 

10.206744 

9 

52 

2588 

3-39 

9050 

3538 

4.69 

6462 

8 

53 

2791 

3-38 

8972 

3819 

4.69 

6181 

7 

54 

2994 

3-38 

8893 

4101 

4.69 

5899 

6 

55 

3197 

3-38 

8815 

4383 

4.69 

5617 

5 

56 

3400 

3.38 

8736 

4664 

4.69 

5336   4 

57 

36°3 

3-37 

8657 

4945 

4.69 

5°55   3 

58 

3805 

3-37 

8578 

5227 

4.69 

4773!  2 

59 

4007 

3-37 

8499 

X-3X 

55°8 

4.68 

449*   1 

60 

9.724210 

9.928420 

9.795789 

10.204211   0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  V 

Tang.     M. 

121°                                      58° 

32°            LOGARITHMIC           147° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l' 

Tang. 

Diff.  1" 

Cotang. 

~o 

9.724210 

3-37 

9.928420 

1.32 

9.795789 

4.68 

10.204211 

60 

1 

4412 

3-37 

8342 

6070 

4.68 

393C 

59 

2 

461; 

3-36 

8263 

635I 

4.68 

3649 

58 

3 

4816 

3.36 

iig; 

6632 

4.68 

3368 

57 

4 

5017 

3-36 

810; 

6913 

4.68 

3087 

56 

5 

5219 

3-36 

8025 

7194 

4.68 

2806 

55 

6 

5420 

3-35 

7946 

7475 

4-68 

2525 

54 

7     5622 

3-35 

7867 

7755 

4.68 

2245 

53 

8     5823 

3-35 

7787 

8036 

4.67 

1964 

52 

9     6024 

3-35 

7708 

8316 

4.67 

1684 

51 

10 

6225 

3-35 

7629 

8596 

4.67 

1404 

50 

11 

9.726426 

3-34 

9.927549 

1.32 

9.798877 

4.67 

IO.2OI  123 

49 

12 

6626 

3-34 

7470 

'•33 

9157 

4.67 

0843 

48 

13 

6827 

3-34 

739° 

9437 

4.67 

056-: 

47 

14 

7027 

3-34 

7310 

9717 

4.67 

028c 

46 

15 

7228 

3-34 

7231 

9-799997 

4.66 

I0.20000C 

45 

16 

7428 

3-33 

7151 

9.800277 

4.66 

10.199723 

44 

17 

7628 

3-33 

7071 

°557 

4.66 

944; 

43 

18 

7828 

3-33 

6991 

0836 

4.66 

9164 

42 

19 

8027 

3-33 

6911 

1116 

4.66 

8884 

41 

20 

8227 

3-33 

6831 

1396 

4.66 

8604 

40 

21 

9.728427 

3-32 

9.926751 

9.801675 

4.66 

10.198325 

39 

22 

8626 

3-32 

6671 

J955 

4.66 

8045 

38 

23 

8825 

3-32 

6591 

'•33 

2234 

4-65 

7766 

37 

24 

9024 

3-3* 

6511 

1.34 

^513 

4.65 

7487 

36 

25 

9223 

3-31 

6431 

2792 

4.65 

7208 

35 

26 

9422 

3-31 

635* 

3072 

4.65 

6928 

34 

27 
28 

9621 
9.729820 

3-3i 
3-31 

6270 
6190 

335i 
3630 

4.65 
4.65 

6649 
6370 

33 
32 

29 
30 

9.730018 
0216 

3-3° 
3-3° 

6no 
6029 

3908 
4187 

4-65 
4.65 

6092 
58i3 

31 

30 

31 

0415 

3-3° 

9.925949 

9.804466 

4.64 

10.195534 

29 

32 

0613 

3-30 

5868 

4745 

4.64 

5255 

28 

33 

0811 

3-3° 

5788 

5023 

4.64 

4977 

27 

34 

1009 

3-29 

57°7 

5302 

4.64 

4698 

26 

35 

1206 

3-29 

5626 

i-34 

558o 

4.64 

4420 

25 

36 

1404 

3-29 

5545 

'•35 

5859 

4.64 

4141 

24 

37 

1602 

3-29 

5465 

6137 

4.64 

3863 

23 

38 

1799 

3-29 

5384 

6415 

4.63 

3585 

22 

39 

1996 

3.28 

53°3 

6693 

4-63 

33°7 

21 

40 

2193 

3.28 

5222 

6971 

4.63 

3029 

20 

41 

9.732390 

3.28 

9.925141 

9.807249 

4-63 

10.192751 

19 

42 

2587 

3.28 

5060 

7527 

4-63 

2473 

18 

43 

2784 

3.28 

4979 

7805 

4-63 

2195 

17 

44 

2980 

3-27 

4897 

8083 

4-63 

1917 

16 

45 

3J77 

3-27 

4816 

i-35 

8361 

4.63 

1639 

15 

46 

3373 

3-27 

4735 

1.36 

8638 

4.62 

1362 

14 

47 

3569 

3-27 

4654 

8916 

4.62 

1084 

13 

48 

3765 

3-27 

4572 

9*93 

4.62 

0807 

12 

49 

3961 

3.26 

4491 

947i 

4.62 

0529 

11 

50 

4i57 

3.26 

4409 

9.809748 

4.62 

10.190252 

10 

51 

9-734353 

3.26 

9.924328 

9.810025 

4.62 

10.189975 

9 

52 

4549 

3.26 

4246 

0302 

4.62 

9698 

8 

53 

4744 

3-25 

4164 

0580 

4.62 

9420 

7 

54 

4939 

3-25 

4083 

0857 

4.62 

9143 

6 

55 

5135 

3-25 

4001 

1134 

4.61 

8866 

5 

56 

533° 

3-25 

39'9 

1410 

4.61 

8590 

4 

57 

5525 

3-25 

3837 

1.36 

1687 

4.61 

8313 

3 

I  58 

5719 

3-24 

3755 

i-37 

1964 

4.61 

8036 

2 

1  59 

59J4 

3-24 

3673 

1.37 

2241 

4.61 

7759 

1 

!  60 

9.736109 

9.923591 

9.812517 

10.187483 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  I" 

Tang. 

M. 

122°                                     57° 

74 


33°       SINES  AlfD  TANGENTS.      146° 

M. 

Sine. 

Diff.  I" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

~T 
i 

9.736109 
6303 

3.24 
3.24 

9.923591 
3509 

1.37 

9.812517 
2794 

4.61 
4.61 

10.187483 
7206 

60 
59 

2 

6498 

3-24 

34^7 

3070 

4.61 

6930 

58 

3 

6692 

3-23 

3345 

3347 

4.60 

6653 

57 

4 

6886 

3-23 

3263 

3623 

4.60 

6377 

56 

5 

7080 

3-23 

3181 

3899 

4.60 

6101 

55 

6 

7274 

3-*3 

3098 

4'75 

4.60 

5825 

54 

7 

7467 

3-*3 

3016 

445  2 

4.60 

5548 

53 

8 

7661 

3.22 

2933 

4728 

4.60 

5272 

52 

9 

7855 

3.22 

2851 

i-37 

5004 

4.60 

4996 

51 

10 

8048 

3.22 

2768 

1.38 

5279 

4.60 

4721 

50 

11 

9.738241 

3.22 

9.922686 

9-8I5555 

4-59 

10.184445 

49 

12 

8434 

3.22 

2603 

5831 

4-59 

4169 

48 

13 

8627 

3.21 

2520 

6107 

4-59 

3893 

47 

14 

8820 

3.21 

2438 

6382 

4-59 

3618 

46 

15 

9013 

3.21 

*355 

6658 

4-59 

3342 

45 

16 

9206 

3.21 

2272 

6933 

4-59 

3067 

44 

17 

9398 

3.21 

2189 

7209 

4-59 

2791 

43 

18 

9590 

3.20 

2106 

7484 

4-59 

2516 

42 

19 

9783 

3.20 

2023 

7759 

4-59 

2241 

41 

20 

9-739975 

3.20 

1940 

1.38 

8035 

4.58 

1965 

40 

21 

9.740167 

3.20 

9.921857 

i-39 

9.818310 

4.58 

10.181690 

39 

22 

0359 

3.20 

1774 

8585 

4.58 

HI5 

38 

23 

0550 

3-*9 

1691 

8860 

4.58 

1140 

37 

24 

0742 

3-19 

1607 

9r35 

4.58 

0865 

36 

25 

0934 

3-19 

1524 

9410 

4-58 

0590 

35 

26 

1125 

3-'9 

1441 

9684 

4.58 

0316 

34 

27 

1316 

3.19 

1357 

9.819959 

4.58 

10.180041 

33 

28 

1508 

3.18 

.   1274 

9.820234 

4-58 

10.179766 

32 

29 

1699 

3.18 

1190 

0508 

4-57 

9492 

31 

30 

1889 

3-i8 

1107 

0783 

4-57 

9217 

30 

31 

9.742080 

3.18 

9.921023 

i-39 

9.821057 

4-57 

10.178943 

29 

32 

2271 

3.18 

0939 

1.40 

1332 

4-57 

8668 

28 

33 

2462 

3-17 

0856 

1606 

4-57 

8394 

27 

34 

2652 

3-*7 

0772 

1880 

4-57 

8120 

26 

35 

2842 

3-i7 

0688 

2154 

4-57 

7846 

25 

36 

3°33 

3-J7 

0604 

2429 

4-57 

7571 

24 

37 

3223 

3-J7 

0520 

2703 

4-57 

7297 

23 

38 

34J3 

3.16 

0436 

2977 

4.56 

7023 

22 

39 

3602 

3.16 

0352 

3250 

4.56 

6750 

21 

40 

379* 

3.16 

0268 

3524 

4.56 

6476 

20 

41 

9.743982 

3.16 

9.920184 

9.823798 

4.56 

10.176202 

19 

42 

4171 

3.16 

0099 

4072 

4.56 

5928 

18 

43 

4361 

3-*5 

9.920015 

1.40 

4345 

4.56 

5655 

17 

44 

455° 

S-'S 

9.919931 

1.41 

4619 

4.56 

538i 

16 

45 

4739 

3-!5 

9846 

4893 

4.56 

5107 

15 

46 

4928 

3-'5 

9762 

5166 

4.56 

4834 

14 

47 

5"7 

3-'5 

9677 

5439 

4-55 

4561 

13 

48 

5306 

3-H 

9593 

57i3 

4-55 

4287 

12 

49 

5494 

3-'4 

9508 

5986 

4-55 

4014 

11 

50 

5683 

3-J4 

9424 

6259 

4-55 

3741 

10 

51 

9.745871 

3-H 

9.919339 

9.826532 

4-55 

10.173468 

9 

52 

6059 

3-H 

9254 

6805 

4-55 

3195 

8 

53 

6248 

3-'3 

9169 

7078 

4-55 

2922 

7 

54 

6436 

3-'3 

9085 

1.41 

735i 

4-55 

2649 

6 

55 

6624 

3-'3 

9000 

1.42 

7624 

4-55 

2376 

5 

56 

6812 

3-'3 

8915 

7897 

4-54 

2103 

4 

57 

6999 

3-J3 

8830 

8170 

4-54 

1830 

3 

58 

7187 

3.12 

8745 

8442 

4-54 

1558 

2 

59 

7374 

3.12 

8659 

1.42 

8715 

4-54 

1285 

1 

60 

9.747562 

9.918574 

9.828987 

10.171013 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

^123°                                       56° 

75 


34°            XiOCtA 

145° 

M. 

Sine. 

Diff.l" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

o 

9.747562 

3.12 

9.918574 

1.42 

9.828987 

4-54 

10.171013 

60 

1 

7749 

3.12 

8489 

9260 

4-54 

0740 

59 

2 

7936 

3.12 

8404 

953* 

4-54 

0468 

58 

3 

4 

8123 
8310 

3.II 
3-11 

8318 
8*33 

9.829805 
9.830077 

4-54 
4-54 

10.170195 
10.169923 

57 
56 

5 

8497 

3.11 

8147 

I.42 

0349 

4-53 

9651 

55 

6 

8683 

3-" 

8062 

1-43 

0621 

4-53 

9379 

54 

7 

8870 

3-" 

7976 

0893 

4-53 

9107 

53 

8 

9056 

3.10 

7891 

1165 

4-53 

8835 

52 

9 

3.10 

7805 

H37 

4-53 

8563 

51 

10 

9429 

3.10 

7719 

1709 

4-53 

8291 

50 

11 

9.749615 

3.10 

9.917634 

9.831981 

4-53 

10.168019 

49 

12 
13 

9801 
9-749987 

3.10 
3-°9 

7548 
7462 

**53 

4-53 
4-53 

7747 
7475 

48 
47 

14 

9.750172 

3-°9 

7376 

2796 

4-53 

7204 

46 

15 

0358 

3-°9 

7290 

3068 

4-5* 

6932 

45 

16 

0543 

3-°9 

7204 

1.43 

3339 

4-5* 

6661 

44 

17 

0729 

3-°9 

7118 

1.44 

3611 

4-5* 

6389 

43 

18 

0914 

3.08 

7032 

3882 

4-5* 

6118 

42 

19 

1099 

3.08 

6946 

4154 

4-5* 

5846 

41 

20 

1284 

3.08 

6859 

44*5 

4-5* 

5575 

40 

~21 

9.751469 

3.08 

9.916773 

9.834696 

4-5* 

10.165304 

39 

22 

1654 

3.08 

6687 

4967 

4-5* 

5°33 

38 

23 

1839 

3.08 

6600 

5*38 

4-5* 

4762 

37 

24 

2027 

3.07 

6514 

55°9 

4-5* 

4491 

36 

25 

2208 

3-°7 

6427 

5780 

4.51 

4220 

35 

26 

2392 

3-°7 

6341 

6051 

4.51 

3949 

34 

27 

2576 

3-°7 

6254 

1.44 

6322 

4.51 

3678 

33 

28 

2760 

3-°7 

6167 

1.45 

6593 

4.51 

3407 

32 

29 

2944 

3.06 

6081 

6864 

4-5  1 

3136 

31 

30 

3128 

3.06 

5994 

7134 

4-5i 

2866 

30 

31 
32 

9-75331* 

3495 

3.06 
3.06 

9.915907 
5820 

9.837405 
7675 

4-1  1 

10.162595 
2325 

29 
28 

33 
34 

3679 
3862 

3.06 
3-°5 

5733 
5646 

7946 
8216 

4-5° 

2054 
1784 

27 
26 

35 

4046 

3-°5 

5559 

8487 

4.50 

1513 

25 

36 

4229 

3.05 

5472 

«757 

4.50 

1*43 

24 

37 

4412 

3-05 

5385 

9027 

4.50 

0973 

23 

38 

4595 

3-°5 

5*97 

9*97 

4.50 

0703 

22 

39 
40 

4778 
4960 

3.04 
3.04 

5210 
5123 

1.45 
1.46 

9568 
9-839838 

4.50 
4.50 

0432 
10.160162 

21 
20 

IT 

9-755H3 

3.04 

9-9I5°35 

9.840108 

4-5° 

10.159892 

19 

42 

5326 

3-°4 

4948 

0378 

4.50 

9622 

18 

43 

5508 

3.04 

4860 

0647 

4.50 

9353 

17 

44 

5690 

3.04 

4773 

0917 

4-49 

9083 

16 

45 

5872 

3-°3 

4685 

1187 

4-49 

8813 

15 

46 

6054 

3-03 

4598 

J457 

4.49 

8543 

14 

47 

6236 

3-°3 

4510 

1726 

4-49 

8274 

13 

48 

6418 

3-03 

4422 

1996 

4-49 

8004 

12 

49 

6600 

3-°3 

4334 

1.46 

2266 

4-49 

7734 

11 

50 

6782 

3.02 

4246 

1.47 

*535 

4-49 

7465 

10 

51 

9.756963 

3.02 

9.914158 

9.842805 

4-49 

10.157195 

9 

52 

7144 

3.02 

4070 

3°74 

4-49 

6926 

8 

53 

7326 

3.02 

3982 

3343 

4-49 

6657 

7 

54 

7507 

3.02 

3612 

4-49 

6388 

6 

55 

7688 

3.01 

3806 

3882 

4.48 

6118 

5 

56 

7869 

3-oi 

3718 

4I51 

4.48 

5849 

4 

57 

8050 

3.01 

3630 

4420 

4.48 

3 

58 

8230 

3.01 

3541 

4689 

4.48 

5311 

2 

59 

8411 

3.01 

3453 

1.47 

4958 

4.48 

5042 

1 

60 

9.758591 

9.913365 

9.845227 

10.154773 

0 

CoKine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

~MT 

124°                                       55° 

76 


35°       SINES  AND  TANGENTS.       144°  • 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9-758591 

3.01 

9.913365 

1.47 

9.845227 

4.48 

10.154773 

60 

1 

8772 

3.00 

3276 

1.47 

5496 

4.48 

4504 

59 

2 

8952 

3.00 

3^7 

1.48 

5764 

4.48 

4236 

58 

3 

9132 

3.00 

3099 

6033 

4.48 

3967 

57 

4 

9312 

3.00 

3010 

6302 

4.48 

3698 

56 

5 

9492 

3.00 

2922 

6570 

4-47 

343° 

55 

6 

9672 

2.99 

2833 

6839 

4-47 

3161 

54 

7 

9.759852 

2.99 

2744 

7107 

4-47 

2893 

53 

8 

9.760031 

2.99 

2655 

7376 

4-47 

2624 

52 

9 

021  1 

2.99 

2566 

7644 

4-47 

2356 

51 

10 

0390 

2.99 

2477 

79'3 

4-47 

2087 

50 

11 

9.760569 

2.98 

9.912388 

1.48 

9.848181 

4-47 

10.151819 

49 

12 

0748 

2.98 

2299 

1.49 

8449 

4-47 

i55i 

48 

13 

0927 

2.98 

22IO 

8717 

4-47 

1283 

47 

14 

1106 

2.98 

2121 

8986 

4-47 

1014 

46 

15 

1285 

2.98 

2031 

9254 

4-47 

0746 

45 

16 

1464 

a.98 

1942 

9522 

4-47 

0478 

44 

17 

1642 

2.97 

1853 

9.849790 

4.46 

10.150210 

43 

18 

1821 

2.97 

1763 

9.850058 

4.46 

10.149942 

42 

19 

1999 

2.97 

1674 

0325 

4.46 

9675 

41 

20 

2177 

2.97 

1584 

°593 

4.46 

9407 

40 

21 
22 

9.762356 
2534 

2.96 

9.9II495 

1405 

1.49 

9.850861 
1129 

4.46 
4.46 

10.149139 

8871 

39 

38 

23 

2712 

2.96 

1315 

1.50 

1396 

4.46 

8604 

37 

24 

2889 

2.96 

1226 

1664 

4.46 

8336 

36 

25 

3067 

2.96 

1136 

I93i 

4.46 

8069 

35 

26 

3*45 

2.96 

1046 

2199 

4.46 

7801 

34 

27 

3422 

2.96 

0956 

2466 

4.46 

7534 

33 

28 

3600 

2.95 

0866 

*733 

4-45 

7267 

32 

29 

3777 

2.95 

0776 

3001 

4-45 

6999 

31 

30 

3954 

2.95 

0686 

3268 

4-45 

6732 

30 

~31 

9.764131 

2.95 

9.910596 

9-853535 

4-45 

10.146465 

29 

32 

4308 

2.95 

0506 

1.50 

3802 

4-45 

6198 

28 

33 
34 

4662 

2.94 
2.94 

WS 
°3^5 

I.5I 

4069 
4336 

4-45 
4-45 

snl 
5664 

27 
26 

35 

4838 

2.94 

0235 

4603 

4-45 

5397 

25 

36 

5°i5 

2.94 

0144 

4870 

4-45 

5^° 

24 

37 

5191 

2.94 

9.910054 

5137 

4-45 

4863 

23 

38 

5367 

2.94 

9.909963 

5404 

4-45 

4596 

22 

39 

5544 

2.93 

9873 

5671 

4-44 

43*9 

21 

40 

5720 

2.93 

9782 

5938 

4.44 

4062 

20 

41 

9.765896 

2.93 

9.90969! 

9.856204 

4.44 

10.143796 

19 

42 

6072 

2.93 

9601 

6471 

4.44 

3529 

18 

43 

6247 

2.93 

9510 

6737 

4.44 

3263 

17 

44 

6423 

2-93 

9419 

1.51 

7004 

4.44 

2996 

16 

45 

6598 

2.92 

9328 

1.52 

7270 

4-44 

2730 

15 

46 

6774 

2.92 

9237 

7537 

4.44 

2463 

14 

47 

6949 

2.92 

9146 

7803 

4.44 

2197 

13 

48 

7124 

2.92 

9°55 

8069 

4.44 

I9JI 

12 

49 

7300 

2.92 

8964 

8336 

4.44 

1664 

11 

50 

7475 

2.91 

8873 

8602 

4-43 

1398 

10 

51 

9.767649 

2.91 

9.908781 

9.858868 

4-43 

10.141132 

9 

52 

7824 

2.91 

8690 

9134 

4-43 

0866 

8 

53 

7999 

2.91 

8599 

9400 

4-43 

0600 

7 

54 

8177 

2.91 

8507 

1.52 

9666 

4-43 

°334 

6 

55 

8348 

2.90 

8416 

'•53 

9.859932 

4-43 

10.140068 

5 

56 

8522 

2.90 

8324 

9.860198 

4-43 

10.139802 

4 

57 

8697 

2.90 

8^33 

0464 

4-43 

9536 

3 

58 

8871 

2.90 

8141 

0730 

4-43 

9270 

2 

59 

9045 

2.90 

8049 

i-53 

0995 

4-43 

9005 

1 

60 

9.769219 

9.907958 

9.861261 

10.138739 

0 

Cosine.    Diff.  I" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

125°                                       64° 

77 


36°            IiOGARJTHXttJC           143° 

M. 

Sine.     Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  I" 

Cotang. 

~0^ 

9.769219     2.90 

9.907958 

1-53 

9.861261 

4-43 

10.138739 

60 

1 

9393    *-89 

7866 

1527 

4-43 

8471 

59 

2 

9566    2.89 

7774 

1792 

4.42 

8208 

58 

3 

9740    2.89 

7682 

2058 

4.42 

7942 

57 

4 
5 

9.769913!    2.89 
9.770087    2.89 

759° 
749  8 

2323 
2589 

4.42 
4.42 

7677 
7411 

56 
55 

6 

0260    2.88 

7406 

1-53 

2854 

4.42 

7146 

54 

7 
8 

0606 

2.88 
2.88 

73H 
7222 

1.54 

3"9 

3385 

4.42 
4.42 

6881 
6615 

53 
52 

9 

0779 

2.88 

7129 

365o 

4.42 

6350 

51 

10 

0952 

2.88 

7037 

39!5 

4.42 

6085 

50 

IT 

9.771125 

2.88 

9.906945 

9.864180 

4.42 

10.135820 

49 

12 

1298 

2.88 

6852 

4445 

4.42 

5555 

48 

13 

1470 

2.87 

6760 

4710 

4.42 

5290 

47 

14 

1643 

2.87 

6667 

4975 

4.41 

5°25 

46 

15 

1815 

2.87 

6575 

5240 

4.41 

4760 

45 

16 

1987 

2.87 

6482 

1.54 

5505 

4.41 

4495 

44 

17 

2159 

2.87 

6389 

i-55 

5770 

4.41 

4230 

43 

18 

2331 

2.86 

6296 

6035 

4.41 

3965 

42 

19 

2503 

2.86 

6204 

6300 

4.41 

3700 

41 

20 

2675 

2.86 

6m 

6564 

4.41 

3436 

40 

~2l 

9.772847 

2.86 

9.906018 

9.866829 

4.41 

10.133171 

39 

22 

3018 

2.86 

5925 

7094 

4.41 

2906 

38 

23 

3190 

2.86 

5832 

7358 

4.41 

2642 

37 

24 

3361 

2.85 

5739 

7623 

4.41 

2377 

36 

25 

3533 

2.85 

5645 

7887 

4.41 

2113 

35 

26 

37°4 

2.85 

5552 

8152 

4.40 

1848 

34 

27 

3875 

2.85 

5459 

i-55 

8416 

4.40 

1584 

33 

28 

4046 

2.85 

5366 

1.56 

8680 

4.40 

1320 

32 

29 

42oZ 

2.85 

5272 

8945 

4.40 

1055 

31 

30 

4388 

2.84 

5179 

9209 

4.40 

0791 

30 

81 

9-774558 

2.84 

9.905085 

9473 

4.40 

0527 

29 

32 

4729 

2.84 

4992 

9.869737 

4.40 

10.130263 

28 

33 

4899 

2.84 

4898 

9.870001 

4.40 

10.129999 

27 

34 

5070 

2.84 

4804 

0265 

4.40 

9735 

26 

35 

5240 

2.84 

4711 

0529 

4.40 

947i 

25 

36 

5410 

2.83 

4617 

0793 

4.40 

9207 

24 

37 

558o 

2.83 

45*3 

1.56 

i°57 

4.40 

8943 

23 

38 

575° 

2.83 

4429 

i-57 

1321 

4.40 

8679 

22 

39 

5920 

2.83 

4335 

1585 

4.40 

8415 

21 

40 

6090 

2.83 

4241 

1849 

4-39 

8151 

20 

~±i 

9.776259 

283 

9.904147 

9.872112 

4-39 

10.127888 

19 

42 

6429 

2.82 

4°53 

2376 

4-39 

7624 

18 

43 

6598 

2.82 

3959 

2640 

4-39 

7360 

17 

44 

6768 

2.82 

3864 

2903 

4-39 

7097 

16 

45 

6937 

2.82 

3770 

3167 

4-39 

6833 

15 

46 

7106 

2.82 

3676 

343° 

4-39 

6570 

14 

47 

7275 

2.8  1 

358i 

3694 

4-39 

6306 

13 

48 

7444 

2.81 

3487 

i-57 

3957 

4-39 

6043 

12 

49 

7613 

2.81 

3392 

1.58 

4220 

4-39 

578o 

11 

50 

7781 

2.81 

3298 

4484 

4-39 

55i6 

10 

61 

9.777950 

2.81 

9.903203 

9.874747 

4-39 

10.125253 

9 

52 

8119 

2.81 

3108 

5010 

4-39 

4990 

8 

53 

8287 

2.80 

3014 

5*73 

4-38 

4727 

7 

54 

8455 

2.80 

2919 

5536 

4.38 

4464 

6 

55 

8624 

2.80 

2824 

5800 

4-38 

4200 

5 

56 

57 

8792 
8960 

2.80 
2.80 

2729 
2634 

6063 
6326 

4-38 
4-38 

3937 
3674 

4 
3 

58 

9128 

2.80 

*539 

1.58 

6589 

4-38 

34" 

2 

59 

9295 

2.79 

2444 

1.59 

6851 

4.38 

3H9 

1 

60 

9.779463 

9.902349 

9.877114 

10.122886 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  I" 

Tang. 

M. 

126°                                       53° 

78 


37°       SI!f  E3  AND  TANGENTS.       142° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.779463 

2.79 

9.902349 

1.59 

9.877114 

4-38 

10.122886 

60 

1 
2 

9631 
9798 

2.79 
2.79 

2253 
2158 

7377 
7640 

4.38 
4-38 

2623 
2360 

59 

58 

3 

9.779966 

2.79 

2063 

79°3 

4-38 

2097 

57 

4 

9.780133 

2.79 

1967 

8165 

4-38 

1835 

56 

5 

0300 

2.78 

1872 

8428 

4-38 

1572 

55 

6 

0467 

2.78 

I776 

8691 

4.38 

1309 

54 

7 

0634 

2.78 

1681 

8953 

4-37 

1047 

53 

8 

0801 

2.78 

I585 

9216 

4-37 

0784 

52 

9 

10 

0968 
"34 

2.78 
2.78 

1490 
1394 

1.59 
1.  60 

9478 
9.879741 

4-37 
4-37 

0522 
10.120259 

51 
50 

11 

9.781301 

2.77 

9.901298 

9.880003 

4-37 

10.119997 

49 

12 

1468 

2.77 

1  202 

0265 

4-37 

9735 

48 

13 

1634 

2.77 

1106 

0528 

4-37 

9472 

47 

14 

18,00 

2.77 

1010 

0790 

4-37 

9210 

46 

15 

1966 

2.77 

0914 

1052 

4-37 

8948 

45 

16 

2132 

2.77 

0818 

I3'4 

4-37 

8686 

44 

17 

2298 

2.76 

0722 

1576 

4-37 

8424 

43 

18 

2464 

2.76 

0626 

l839 

4-37 

8161 

42 

19 

2630 

2.76 

0529 

1.  60 

2101 

4-37 

7899 

41 

20 

2796 

2.76 

°433 

1.61 

2363 

4.36 

7637 

40 

21 

9.782961 

2.76 

9.900337 

9.882625 

4.36 

10.117375 

39 

22 

3127 

2.76 

0240 

2887 

4.36 

?o13 

38 

23 

3292 

2.75 

0144 

3148 

4.36 

6852 

37 

24 

3458 

2.75 

9.900047 

3410 

4.36 

6590 

36 

25 

3623 

2.75 

9.899951 

3672 

4.36 

6328 

35 

26 

3788 

2.75 

9854 

3934 

4.36 

6066 

34 

27 

3953 

2.75 

9757 

4196 

4.36 

5804 

33 

28 

4118 

2.74 

9660 

4457 

4-36 

5543 

32 

29 

4282 

2.74 

9564 

1.61 

47i9 

4.36 

5281 

31 

30 

4447 

2.74 

9467 

1.62 

4980 

4.36 

5020 

30 

~3l 

9.784612 

2.74 

9.899370 

9.885242 

4-36 

10.114758 

29 

32 

4776 

2.74 

9273 

55°3 

4.36 

4497 

28 

8? 

4941 

2.74 

9176 

5765 

4-36 

4235 

27 

34 

5I05 

2.74 

9078 

6026 

4-36 

3974 

26 

35 

5269 

2.73 

8981 

6288 

4-36 

3712 

25 

36 

5433 

2-73 

8884 

6549 

4-35 

345  i 

24 

37 

5597 

a-73 

8787 

6810 

4-35 

3190 

23 

38 

576i 

a-73 

8689 

7072 

4-35 

2928 

22 

39 

5925 

2.73 

8592 

1.62 

7333 

4-35 

2667 

21 

40 

6089 

2-73 

8494 

1.63 

7594 

4-35 

2406 

20 

41 

9.786252 

2.72 

9.898397 

9.887855 

4-35 

10.112145 

19 

42 

6416 

2.72 

8299 

8116 

4-35 

1884 

18 

43 

6579 

2.72 

8202 

8377 

4-35 

1623 

17 

44 

6742 

2.72 

8104 

8639 

4-35 

1361 

16 

45 

6906 

2.72 

8006 

8900 

4-35 

1  100 

15 

46 

7069 

2.72 

7908 

9160 

4-35 

0840 

14 

47 

7232 

2.71 

7810 

9421 

4-35 

0579 

13 

48 

7395 

2.71 

7712 

9682 

4-35 

0318 

12 

49 

7557 

2.71 

7614 

9.889943 

4-35 

10.110057 

11 

50 

7720 

2.71 

7516 

1.63 

9.890204 

4-34 

10.109796 

10 

51 

9.787883 

2.71 

9.897418 

1.64 

0465 

4-34 

9535 

"V 

52 

8045 

2.71 

7320 

0725 

4-34 

9275 

8 

53 

8208 

2.71 

7222 

0986 

4-34 

9014 

7 

54 

8370 

2.70 

7123 

1247 

4-34 

8753 

6 

55 

8532 

2.70 

7025 

J5°7 

4-34 

8493 

5 

56 

8694 

2.70 

6926 

1768 

4-34 

8232 

4 

57 

8856 

2.70 

6828 

2028 

4-34 

7972 

3 

58 

9018 

2.70 

6729 

2289 

4-34 

7711 

2 

59 

9180 

2.70 

6631 

1.64 

2549 

4-34 

745  * 

1 

60 

9.789342 

9.896532 

9.892810 

10.107190 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  I" 

Tang. 

M. 

127°                                      52° 

79 


38°            LOGARITHMIC            141° 

M. 

Sine. 

Diff.l" 

Cosine. 

Diff.l' 

Tang. 

Diff.  1" 

Cotang.   I 

~0 

9.789342 

2.69 

9.896532 

1.64 

9.892810 

4-34 

10.107190  60 

1 

9504 

2.69 

6433 

1.65 

3070 

4-34 

6930  59 

2 

9665 

2.69 

6335 

3331 

4-34 

6669  58 

3 
4 

9827 
9.789988 

2.69 
2.69 

6236 
6137 

Ss! 

4-34 
4-34 

6409 
6149 

57 
56 

5 

9.790149 

2.69 

6038 

4111 

4-34 

5889 

55 

6 

7 

0310 
0471 

2.68 
2.68 

5939 

5840 

4371 
4632 

4-34 
4-33 

5629 

54 
53 

8 

0632 

2.68 

4892 

4-33 

5108 

52 

9 

°793 

2.68 

5641 

4-33 

4848 

51 

10 

°954 

2.68 

554* 

I.6S 

54^ 

4-33 

4588 

50 

11 

9.791115 

2.68 

9.895443 

1.66 

9.895672 

4-33 

10.104328 

49 

12 

1275 

2.67 

5343 

5932 

4-33 

4068 

48 

13 

I436 

2.67 

5*44 

6192 

4-33 

3808 

47 

14 

1596 

2.67 

5H5 

6452 

4-33 

3548 

46 

15 

1757 

2.67 

5°45 

6712 

4-33 

3288 

45 

16 

1917 

2.67 

4945 

6971 

4-33 

3029 

44 

17 

2077 

2.67 

4846 

7231 

4-33 

2769 

43 

18 

2237 

2.66 

4746 

749  1 

4-33 

2509 

42 

19 

2397 

2.66 

4646 

7751 

4-33 

2249 

41 

20 

*557 

2.66 

4546 

1.66 

8010 

4-33 

1990 

40 

22 

9.792716 

2876 

2.66 
2.66 

9.894446 
4346 

1.67 

9.898270 
8530 

4-33 
4-33 

10.101730 
1470 

39 

38 

23 

3°35 

2.66 

4246 

8789 

13X1 

37 

24 

3195 

2.66 

4146 

9049 

4.32 

0951 

36 

25 

3354 

2.65 

4046 

93°8 

4-3* 

0692 

35 

26 
27 

28 

35H 
3673 
3832 

2.65 
2.65 
2.65 

3946 
3846 
3745 

9568 
9.899827 
9.900086 

4-3* 
4-3* 
4.32 

0432 

10.100173 
10.099914 

34 
33 
32 

29 

3991 

2.65 

3645 

0346 

4-3* 

9654 

31 

30 

4150 

2.64 

3544 

1.67 

0605 

4.32 

9395 

30 

~3T 

9.794308 

2.64 

9.893444 

1.68 

9.900864 

4.32 

10.099136 

29 

32 

2.64 

3343 

1124 

4-32 

8876 

28 

33 

4626 

2.64 

3*43 

1383 

4-32 

8617 

27 

34 

4784 

2.64 

1642 

8358 

26 

35 

4942 

2.64 

3041 

1901 

4-3* 

8099 

25 

36 

5101 

2.64 

2940 

2160 

4-3* 

7840 

24 

37 

5259 

2.63 

2839 

2419 

4.32 

7581 

23 

38 
39 

5417 
5575 

2.63 
2.63 

2739 
2638 

2679 
2938 

4-3* 
4.32 

7321 
7002 

22 
21 

40 

5733 

2.63 

2536 

1.68 

3*97 

4.31 

6803 

20 

41 

9.795891 

2.63 

9.892435 

1.69 

9-9°3455 

4-31 

10.096545 

19 

42 

6049 

2.63 

2334 

37H 

4.31 

6286 

18 

43 

6206 

2.63 

3973 

4.31 

6027 

17 

44 

6364 

2.62 

2132 

4232 

4.31 

5768 

16 

45 

6521 

2.62 

2030 

4491 

4.31 

55°9 

15 

46 

6679 

2.62 

1929 

4750 

4-3  * 

5250 

14 

47 

6836 

2.62 

1827 

5008 

4-3  * 

4992 

13 

48 

6993 

2.62 

1726 

5267 

4-3  * 

4733 

12 

49 

7150 

2.62 

1624 

1.69 

55*6 

4.31 

4474 

11 

50 

7307 

2.61 

1523 

1.70 

5784 

4-31 

4216 

10 

51 

9.797464 

2.61 

9.891421 

9.906043 

4-3  » 

10.093957 

9 

52 

7621 

2.61 

1319 

6302 

4.31 

3698 

8 

53 

7777 

2.61 

1217 

6560 

4.31 

344° 

7 

54 

7934 

2.61 

1115 

6819 

4.31 

3181 

6 

55 

8091 

2.61 

1013 

7077 

4.31 

2923 

5 

56 

8247 

2.6  1 

0911 

7336 

4.31 

2664 

4 

57 

8403 

2.60 

.  0809 

7594 

4.31 

2406 

3 

58 

8560 

2.60 

0707 

7852 

4-31 

2148 

2 

59 

8716 

2.60 

0605 

1.70 

8m 

4.30 

1889 

1 

60 

9.798872 

9.890503 

9.908369 

10.091631 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

If. 

128°                                       51° 

80 


39°       SINES  AND  TANGENTS.       140° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.798872 

2.60 

9.890503 

1.70 

9.908369 

4.30 

10.091631 

60 

1 

9028 

2.60 

0400 

1.71 

8628 

4-3° 

1372 

59 

2 

9184 

2.60 

0298 

8886 

4.30 

:  i  i  ; 

58 

3 

9339 

2.59 

0195 

9144 

4-3° 

0856 

57 

4 

9495 

2.59 

9.890093 

9402 

4.30 

0598 

56 

5 

9651 

2.59 

9.889990 

9660 

4.30 

0340 

55 

6 

9806 

2.59 

9888 

9.909918 

4.30 

10.090082 

54 

7 

9.799962 

2.59 

9785 

9.910177 

4-3° 

10.089823 

53 

8 

9.800117 

2.59 

9682 

°435 

4.30 

9565 

52 

9 

0272 

2.58 

9579 

0693 

4.30 

9307 

51 

10 

0427 

2.58 

9477 

1.71 

0951 

4.30 

9049 

50 

11 

9.800582 

2.58 

9.889374 

1.72 

9.911209 

4-3° 

10.088791 

49 

12 

0737 

2.58 

9271 

1467 

4-3° 

8533 

48 

13 

0892 

2.58 

9168 

1724 

4-3° 

8276 

47 

14 

1047 

2.58 

9064 

1982 

4-3° 

8018 

46 

15 

1201 

2.58 

8961 

2240 

4.30 

7760 

45 

16 

1356 

*-57 

8858 

2498 

4-3° 

7502 

44 

17 

2.57 

8755 

2756 

4.30 

7244 

43 

18 

1665 

2.57 

8651 

3OI4 

4.29 

6986 

42 

19 

1819 

2.57 

8548 

1.72 

3*7i 

4.29 

6729 

41 

20 

1973 

2.57 

8444 

1.73 

35*9 

4.29 

6471 

40 

21 

22 

9.802128 
2282 

2.57 
2.56 

9.888341 

8237 

9.913787 
4044 

4.29 
4.29 

10.086213 
5956 

39 
38 

L>3 

2436 

2.56 

8134 

4302 

4-*9 

5698 

37 

24 

2589 

2.56 

8030 

4560 

4.29 

544° 

36 

25 

2743 

2.56 

7926 

4817 

4.29 

5183 

35 

26 

2897 

2.56 

7822 

5°75 

4.29 

49*5 

34 

27 

3050 

,*.56 

7718 

533* 

4-*9 

4668 

33 

28 

3204 

2.56 

7614 

559° 

4,29 

4410 

32 

29 

3357 

2.55 

7510 

1.73 

5847 

4.29 

4153 

31 

30 

3511 

2.55 

7406 

1.74 

6104 

4.29 

3896 

30 

31 

9.803664 

2.55 

9.887302 

9.916362 

4.29 

10.083638 

29 

32 

3817 

2.55 

7198 

6619 

4.29 

338i 

28 

33 
34 

3970 
4'*3 

2.55 
2.55 

6989 

6877 
7134 

4.29 
4.29 

mi 

27 
26 

35 

4276 

2.54 

6885 

7391 

4.29 

2609 

25 

36 

4428 

a.  54 

6780 

7648 

4.29 

2352 

24 

37 

458i 

*-54 

6676 

7905 

4.29 

2095 

23 

38 

2.54 

6571 

8163 

4.28 

1837 

22 

39 

4886 

2.54 

6466 

1.74 

8420 

4.28 

1580 

21 

40 

5039 

2.54 

6362 

J-75 

8677 

4.28 

1323 

20 

41 

9.805191 

2.54 

9.886257 

9.918934 

4.28 

10.081066 

19 

42 

5343 

2.53 

6152 

9191 

4.28 

0809 

18 

43 

5495 

2.53 

6047 

9448 

4.28 

0552 

17 

44 

5647 

2,53 

5942 

9705 

4.28 

029C 

16 

45 

5799 

4.53 

5837 

9.919962 

4.28 

10.080038 

15 

46 

5951 

*-53 

573* 

9.920219 

4.28 

10.079781 

14 

47 

6103 

2.53 

5627 

0476 

4.28 

95*4 

13 

48 

6254 

2.53 

55** 

0733 

4.28 

9267 

12 

49 
60 

6406 
6557 

2.52 
2.52 

5416 
53" 

'•75 
1.76 

0990 

1*47 

4.28 
4-28 

9010 
8753 

11 
II 

~5T 

.02 

9.806709 
6860 

2.52 
2.52 

9.885205 
5100 

9.921503 
1760 

4.28 
4.28 

10.078497 

82AO 

9 

8 

53 

7011 

2.52 

4994 

2017 

4.28 

7983 

7 

54 

7163 

2.52 

4889 

2274 

4.28 

7726 

6 

55 

73H 

2.52 

4783 

2530 

4.28 

7470 

5 

56 

.07 

7465 
7*766 

2.51 
2.51 
2.51 

4677 

457* 
4466 

1.76 
1.77 

2787 

3°44 
3300 

4.28 
4.28 
4.28 

6956 

6700 

4 
3 
2 

59 

791  7 

2,51 

4360 

1.77 

3557 

4.27 

6443 

1 

GO 

9.808067 

9.884254 

9.923813 

10.076187 

0 

Cosine. 

Diff.  V 

Bine. 

Diff.1" 

Cotang.   Diff.  1" 

Tang. 

M. 

129°                                       60° 

81 


40°        XIOGAR.ITHRXXC       139° 

M.    Sine. 

Diff.  V 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

0 

9.808067 

2.51 

9.884254 

1.77 

9.923813 

4.28 

10.076187 

60 

1 

8218 

2.51 

4148 

4070 

4.27 

593° 

59 

2 

8368 

2.51 

4042 

4327 

4.27 

5673 

58 

3 

8519 

2.50 

3936 

4583 

4.27 

54*7 

57 

4 

8669 

2.50 

3829 

4840 

4.27 

5160 

56 

5 

8819 

2.50 

37^3 

5096 

4.27 

4904 

55 

6 

8969 

2.50 

3617 

5352 

4.27 

4648 

54 

7 

9119 

2.50 

35io 

5609 

4.27 

4391 

53 

8 

9269 

2.50 

3404 

1.77 

5865 

4.27 

4*35 

52 

9 

9419 

2.49 

3*97 

I.78 

6l22 

4.27 

3878 

51 

10 

9569 

2.49 

3191 

6378 

4.27 

3622 

50 

11 

12 

9718 
9.809868 

2.49 
2.49 

9.883084 
2977 

9.926634 
6890 

4.27 
4.27 

10.073366 
3110 

49 

48 

13 

9.810017 

2.49 

2871 

7147 

4.27 

*853 

47 

14 

0167 

2.49 

2764 

7403 

4.27 

2597 

46 

15 

0316 

2.48 

2657 

7659 

4.27 

*34' 

45 

16 

0465 

2.48 

*55° 

79'5 

4.27 

2085 

44 

17 

0614 

2.48 

*443 

I.78 

8171 

4.27 

1829 

43 

18 

0763 

2.48 

2336 

1.79 

8427 

4.27 

1573 

42 

19 

0912 

2.48 

2229 

8683 

4.27 

1317 

41 

21 

1061 

2.48 

2121 

8940 

4.27 

1060 

40 

~2l 

9.811210 

2.48 

9.882014 

9.929196 

4.27 

10.070804 

39 

22 

1358 

2.48 

1907 

9452 

4.27 

0548 

38 

23 

i5°7 

3.47 

1799 

9708 

4.27 

0292 

37 

24 

1655 

2.47 

1692 

9.929964 

4.27 

10.070036 

36 

25 

1804 

2.47 

I584 

9.930220 

4.26 

10.069780 

35 

26 

1952 

2.47 

*477 

°475 

4.26 

952"; 

34 

27 

2100 

2.47 

1369 

1.79 

0731 

4.26      9269 

33 

28 

2248 

2.47 

1261 

1.80 

0987 

|L&6l        0017 

32 

29 

2396 

2.46 

"53 

i*43 

4.26 

8757 

81 

30 

2544 

2.46 

1046 

1499 

4.26 

8501 

30 

31 

9.812692 

2.46 

9.880938 

9-93'755 

4.26 

10.068245 

29 

32 

2840 

2.46 

0830 

2OIO 

4.26 

799° 

28 

33 

2988 

2.46 

0722 

2266 

4.26 

7734 

27 

34 

3135 

2.46 

0613 

2522 

4.26 

7478 

26 

35 

3283 

2.46 

05°5 

2778 

4.26 

7222 

25 

36 

343° 

2.46 

°397 

i.  80 

3°33 

4.26 

6967 

24 

37 

3578 

a-45 

0289 

1.81 

3289 

4.26 

6711 

23 

38 

3725 

2.45 

0180 

3545 

4.26 

6455 

22 

39 

3872 

2.45 

9.880072 

3800 

4.26 

6200 

21 

40 

4019 

2.45 

9.879963 

4056 

4.26 

5944 

20 

41 

9.814166 

2.45 

9855 

9-9343" 

4,26 

10.065689 

19 

42 

4313 

2.45 

9746 

4567 

4.26 

5433 

18 

43 

4460 

2.44 

9637 

4823 

4.26 

S*77 

17 

44 

4607 

2.44 

9529 

5°78 

4.26 

4922 

16 

45 

4753 

2.44 

9420 

5333 

4.26 

4667 

15 

46 

4900 

2.44 

9311 

1.81 

5589 

4.26 

4411 

14 

47 

5046 

2.44 

9202, 

1.82 

5844 

4.26 

4i56 

13 

48 

5193 

2.44 

9°93 

6100 

4.26 

3900 

12 

49 

5339 

2.44 

8984 

6355 

4.26 

3645 

11  ! 

50 

5485 

^•43 

8875 

6610 

4.26 

339° 

10 

"31 

9.815631 

2-43 

9.878766 

9.936866 

4-25 

10.063134 

9 

52 

5778 

2.43 

8656 

7121 

4-25 

2879 

8 

53 

5924 

2-43 

8547 

7376 

4.25 

2624 

7 

54 

6069 

2.43 

8438 

7632 

4-25 

2368 

6 

55 

6215 

2.43 

8328 

1.82 

7887 

4-25 

2113 

5 

56 

6361 

^•43 

8219 

1.83 

8142 

4.25 

1858 

4 

57 

6507 

2.42 

8109 

8398 

4.25 

1602 

3 

58 

6652 

2.42 

7999 

8653 

4.25 

1347 

2 

59 

6798 

2.42 

7890 

1.83 

8908 

4.25 

1092 

I 

60 

9.816943 

9.877780 

9.939163 

10.060837 

0 

Cosine. 

Diff.  V 

Sine. 

Diff.l" 

Cotang. 

Diff.  V 

Tang. 

M. 

130°                                       49° 

41°       SI3KTES  AND  TAZUaZUXTTS.       138°  ] 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  V 

Cotang. 

0 

9.816943 

2.42 

9.877780 

^83 

9.939163 

4-25 

10.060837 

60 

1 

7088 

2.42 

7670 

9418 

4.25 

0582 

59 

2 

7233 

2.42 

7560 

9673 

4.25 

0327 

58 

3 

7379 

2.42 

745° 

9.939928 

4.25 

10.060072 

57 

4 

7524 

2.42 

734° 

1.83 

9.940183 

4.25 

10.059817 

56 

5 

7668 

2.41 

7230 

1.84 

0438 

4.25 

9562 

55 

6 

7813 

2.4I 

7120 

0694 

4-25 

9306 

54 

7 

795* 

2.41 

7010 

0949 

4.25 

9°5! 

53 

8 

8103 

2.4I 

6899 

1204 

4.25 

8796 

52 

9 

8247 

2.41 

6789 

1458 

4-25 

8542 

51 

10 

8392 

2.41 

6678 

1714 

4.25 

8286 

50 

11 

9.818536 

2.40 

9.876568 

9.941968 

4.25 

10.058032 

49 

12 

8681 

2.40 

6457 

2223 

4.25 

7777 

48 

13 

8825 

2.40 

6347 

1.84 

2478 

4.25 

7522 

47 

14 

8969 

2.40 

6236 

1.85 

2733 

4.25 

7267 

46 

15 

9113 

2.40 

6125 

2988 

4.25 

7012 

45 

16 

9257 

2.40 

6014 

3243 

4.25 

6757 

44 

17 

9401 

0,40 

59°4 

3498 

4.25 

6502 

43 

18 

9545 

2.40 

5793 

3752 

4.25 

6248 

42 

19 

9689 

2-39 

5682 

4007 

4.25 

5993 

41 

20 

9832 

2-39 

557i 

4262 

4.25 

5738 

40 

21 

9.819976 

2.39 

9-*75459 

9.944517 

4.25 

10.055483 

39 

22 

9.820120 

2-39 

5348 

477  * 

4.24 

5229 

38 

23 

0263 

2-39 

5237 

1.85 

5026 

4.24 

4974 

37 

24 

0406 

2-39 

5126 

1.86 

5281 

4.24 

47*9 

36 

25 

°55° 

*.38 

5014 

5535 

4.24 

4465 

35 

26 

0693 

*.38 

49°3 

579° 

4.24 

4210 

34 

27 

0836 

2.38 

47  9  * 

6045 

4.24 

3955 

33 

28 

0979 

2.38 

4680 

6299 

4.24 

3701 

32 

29 

1122 

a.38 

4568 

6554 

4.24 

3446 

31 

30 

1265 

2.38 

4456 

6808 

4.24 

3192 

30 

31 

9.821407 

2.38 

9-874344 

i.S6 

9-947063 

4.24 

10.052937 

~29~ 

32 

155° 

2.38 

4232 

1.87 

73i* 

4.24 

2682 

28 

33 

1693 

2-37 

4121 

7572 

4.24 

2428 

27 

34 

1835 

2.37 

4009 

7826 

4-24 

2174 

26 

35 

1977 

2-37 

3896 

8081 

4.24 

1919 

25 

36 

2I2O 

2.37 

3784 

8336 

4.24 

1664 

24 

37 

2262 

2.37 

3672 

8590 

4.24 

1410 

23 

38 

2404 

2-37 

3560 

8844 

4.24 

1156 

22 

39 

2546 

2-37 

3448 

9099 

4.24 

0901 

21 

40 

2688 

2.36 

3335 

9353 

4.24 

0647 

20 

41 

9.822830 

2.36 

9.873223 

1.87 

9607 

4.24 

°393 

19 

42 
43 

2972 

3"4 

2.36 
2.36 

3110 
2998 

1.88 

9.949862 
9.950116 

4.24 
4.24 

10.050138 
10.049884 

18 
17 

44 

3*55 

2.36 

2885 

0370 

4.24 

9630 

16 

45 

3397 

2.36 

2772 

0625 

4.24 

9375 

15 

46 

3539 

2.36 

2659 

0879 

4.24 

9121 

14 

47 

3680 

z-35 

2547 

JI33 

4.24 

8867 

13 

48 

3821 

2-35 

2434 

1388 

4.24 

8612 

12 

49 

3963 

2.35 

2321 

1642 

4.24 

8358 

11 

50 

4104 

2-35 

2208 

1.88 

1896 

4.24 

8104 

10 

61 

9.824245 

2-35 

9.872095 

1.89 

9.952150 

4.24 

10.047850 

9 

52 

4386 

2-35 

1981 

2405 

4.24 

7595 

8 

53 

45*7 

^•35 

1868 

2659 

4.24 

734i 

7 

54 

4668 

2-34 

1755 

2913 

4.24 

7087 

6 

55 

4808 

2-34 

1641 

3167 

4.23 

6833 

5 

56 

4949 

2-34 

1528 

3421 

4-23 

6579 

4 

57 

5090 

2-34 

1414 

3675 

4-23 

63*5 

3 

58 

5230 

2-34 

1301 

3929 

4.23 

6071 

2 

59 

5371 

2-34 

1187 

1.89 

4183 

4.23 

5817 

1 

60 

9.825511 

9.871073 

9-954437 

10.045563 

0 

Cosine. 

Diff.  I" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tun-. 

M. 

131°                                        48° 

83 


42°            ZiOCtARITHimC            137° 

M. 

Sine. 

Diff.  1" 

Cosine. 

Diff.  1" 

Tang. 

Diff.  1" 

Cotang. 

0 

9.825511 

2.34 

9.871073 

I.9o 

9-954437 

4.23 

10.045563 

60 

1 

5651 

2.33 

0960 

4691 

4.23 

53°9 

59 

2 

2-33 

0846 

4945 

5°55 

58 

3 

5931 

2.33 

0732 

5200 

4.23 

4800 

57 

4 

6071 

2-33 

0618 

5454 

4.23 

4546 

56 

5 

6211 

*-33 

0504 

57°7 

4-23 

4293 

55 

6 

6351 

2.33 

0390 

5961 

4-23 

4°39 

54 

7 

6491 

2.33 

0276 

6215 

4-23 

3785 

53 

8 

6631 

2.33 

0161 

1.90 

6469 

52 

0 

6770 

2.32 

9.870047 

1.91 

6723 

4.23 

3277 

51 

10 

6910 

2.32 

9.869933 

6977 

4-23 

3023 

50 

11 

9.827049 

2.32 

9818 

9.957231 

4.23 

10.042769 

49 

12 

7189 

2.32 

9704 

7485 

4.23 

2515 

48 

13 

7328 

2.32 

9589 

7739 

4.23 

2261 

47 

14 

7467 

2.32 

9474 

7993 

4.23 

2007 

46 

15 

7606 

2.32 

9360 

8246 

4-23 

J754 

45 

16 

7745 

2.32 

9245 

8500 

4.23 

1500 

44 

17 

7884 

2.31 

9130 

1.91 

8754 

4-23 

1246 

43 

18 

8023 

2.31 

9015 

1.9-2 

9008 

0992 

42 

19 

8162 

2.31 

8900 

9262 

4.23 

0738 

41 

20 

8301 

2.31 

8785 

9516 

4-23 

0484 

40 

21 

9.828439 

2.31 

9.868670 

9.959769 

4-23 

10.040231 

39 

22 

8578 

2.31 

8555 

9.960023 

4-23 

10.039977 

38 

23 

8716 

2.31 

8440 

0277 

4.23 

9723 

37 

24 

8855 

2.30 

8324 

0531 

4.23 

9469 

36 

25 

8993 

2.30 

8209 

0784 

4.23 

9216 

35 

26 

27 

9*3' 
9269 

2.30 
2.30 

8093 
7978 

1.92 

1038 
1291 

4-23 
4.23 

8962 
8709 

34 
33 

28 

9407 

2.30 

7862 

'545 

4-23 

8455 

32 

29 

9545 

2.30 

7747 

1799 

4.23 

8201 

31 

30 

9683 

2.30 

7631 

2052 

7948 

30 

31 

9821 

2.29 

9.867515 

9.962306 

4.23 

10.037694 

29 

32 

9.829959 

2.29 

7399 

2560 

4.23 

7440 

28 

33 

9.830097 

2.29 

7283 

2813 

4.23 

7187 

27 

34 

0234 

2.29 

7167 

3067 

4.23 

6933 

26 

35 

0372 

2.29 

7051 

I.93 

3320 

4-23 

6680 

25 

36 

0509 

2.29 

6935 

1.94 

3574 

4-23 

6426 

24 

37 

0646 

2.29 

6819 

3827 

4.23 

6173 

23 

38 

0784 

2.29 

6703 

4081 

4.23 

59'9 

22 

39 

0921 

2.28 

6586 

4335 

5665 

21 

40 

1058 

2.28 

6470 

4588 

4.22 

5412 

20 

"If 

9.831195 

2.28 

9.866353 

9.964842 

4.22 

10.035158 

19 

42 

1332 

2.28 

6237 

5°95 

4.22 

4905 

18 

43 

1469 

2.28 

6120 

1.94 

5349 

4.22 

4651 

17 

44 

1606 

2.28 

6004 

1.95 

5602 

4.22 

4398 

16 

45 

1742 

2.28 

5887 

5855 

4.22 

4H5 

15 

46 

1879 

2.28 

5770 

6109 

4.22 

3891 

14 

47 

2015 

2.27 

5653 

6362 

4.22 

3638 

13 

48 

2152 

2.27 

5536 

6616 

4.22 

3384 

12 

49 

2288 

2.27 

5419 

6869 

4.22 

3*3! 

11 

50 

2425 

2.27 

5302 

7123 

4.22 

2877 

10 

51 

9.832561 

2.27 

9.865185 

9.967376 

4.22 

10.032624 

9 

52 

2697 

2.27 

5068 

7629 

4.22 

2371 

8 

53 
54 

2833 
2969 

2.27 
2.26 

4950 
4833 

1.95 
1.96 

7883 
8136 

4.22 
4.22 

2117 
1864 

7 
6 

55 

3I05 

2.26 

4716 

8389 

4.22 

1611 

5  1 

56 

3*41 

2.26 

4598 

8643 

4.22 

1357 

4  1 

57 

3377 

2.26 

4481 

8896 

4.22 

1104 

3 

58 

3512 

2.26 

4363 

9149 

4.22 

0851 

2  ! 

59 

3648 

2.26 

4245 

1.96 

4.22 

0597 

1  I 

60 

9.833783 

9.864127 

9.969656 

10.030344 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.]" 

Cotang. 

Diff.  1" 

Tang. 

M. 

132°                                       47° 

43°       SINES  AND  TANGENTS.      136° 

Jkl. 

Sine. 

Diff.  1" 

Cosine. 

Diff.l" 

Tang. 

Diff.  1" 

Cotang. 

~0 

9^33783 

2.26 

9.864127 

1.96 

9.969656 

4.22 

10.030344 

60 

1 

3919 

2.25 

4010 

1.96 

9909 

4.22 

0091 

59 

2 

4054 

2.25 

3892 

1.97 

9.970162 

4.22 

10.029838 

58 

3 

4189 

2.25 

3774 

0416 

4.22 

9584 

57 

4 

43*5 

2.25 

3656 

0669 

4.22 

9331 

56 

5 

4460 

2.25 

3538 

0922 

4.22 

9078 

55 

6 

4595 

2.25 

3419 

1175 

4.22 

8825 

54 

7 

473° 

2.25 

33°i 

1429 

4.22 

8571 

53 

8 

4865 

2.25 

3183 

1682 

4.22 

8318 

52 

9 

4999 

2.24 

3064 

1.97 

*935 

4.22 

8065 

51 

10 

5*34 

2.24 

2946 

I.98 

2188 

4.22 

7812 

50 

~TT 

9.835269 

2.24 

9.862827 

9.972441 

4.22 

10.027559 

49 

12 

5403 

2.24 

2709 

2694 

4.22 

7306 

48 

13 

5538 

2.24 

2590 

2948 

4.22 

7052 

47 

14 

5672 

2.24 

2471 

3201 

4.22 

6799 

46 

15 

5807 

2.24 

*353 

3454 

4.22 

6546 

45 

16 

594i 

2.24 

2234 

37°7 

4.22 

6293 

44 

17 

6075 

2.23 

2115 

3960 

4.22 

6040 

43 

18 

6209 

2.23 

1996 

4213 

4.22 

5787 

42 

19 

6343 

2.23 

1877 

I.98 

4466 

4.22 

5534 

41 

20 

6477 

2.23 

1758 

1.99 

4719 

4.22 

5281 

40 

21 

9.836611 

2.23 

9.861638 

9-974973 

4.22 

10.025027 

39 

22 

6745 

2.23 

1519 

5226 

4.22 

4774 

38 

23 

6878 

2.23 

1400 

5479 

4.22 

4521 

37 

24 

7012 

2.22 

1280 

573* 

4.22 

4268 

36 

25 

7146 

2.22 

1161 

5985 

4.22 

4015 

35 

26 

7279 

2.22 

1041 

6238 

4.22 

3762 

34 

27 

7412 

2.22 

0922 

6491 

4.22 

35°9 

33 

28 

7546 

2.22 

0802 

1.99 

6744 

4.22 

3256 

32 

29 

7679 

2.22 

0682 

2.00 

6997 

4.22 

3003 

31 

30 

7812 

2.22 

0562 

7250 

4.22 

2750 

30 

31 

9-837945 

2.22 

9.860442 

9-977503 

4.22 

10.022497 

29 

32 

8078 

2.21 

0322 

7756 

4.22 

2244 

28 

33 

8211 

2.21 

O2O2 

8009 

4.22 

1991 

27 

34 

8344 

2.21 

9.860082 

8262 

4.22 

1738 

26 

35 

8477 

2.21 

9.859962 

8515 

4.22 

1485 

25 

36 

8610 

2.21 

9842 

2.00 

8768 

4.22 

1232 

24 

37 

8742 

2.21 

9721 

2.01 

9021 

4.22 

0979 

23 

38 

8875 

2.21 

9601 

9274 

4.22 

0726 

22 

39 

9007 

2.21 

9480 

9527 

4.22 

0473 

21 

40 

9140 

2.20 

9360 

9.979780 

4.22 

IO.O2O22O 

20 

41 

9.839272 

2.20 

9.859239 

9.980033 

4.22 

10.019967 

19 

42 

9404 

2.20 

9119 

0286 

4.22 

9714 

18 

43 

9536 

2.20 

8998 

0538 

4.22 

9462 

17 

44 

9668 

2.20 

8877 

2.01 

0791 

4.21 

9209 

16 

45 

9800 

2.20 

8756 

2.02 

1044 

4.21 

8956 

15 

46 

9.839932 

2.  2O 

8635 

1297 

4.21 

8703 

14 

47 

9.840064 

2.19 

85H 

1550 

4.21 

8450 

13 

48 

0196 

2.19 

8393 

1803 

4.21 

8l97 

12 

49 

0328 

2.19 

8272 

2056 

4.21 

7944 

11 

50 

°459 

2.19 

8151 

2309 

4.21 

7691 

10 

51 

9.840591 

2.19 

9.858029 

9.982562 

4.21 

10.017438 

~9~ 

52 

0722 

2.19 

7908 

2814 

4.21 

7186 

8 

53 

0854 

2.19 

7786 

2.  02 

3067 

4.21 

6933 

7 

54 

55 

0985 
1116 

2.19 
2.19 

7665 

7543 

2.03 

3320 
3573 

4.2I 
4.21 

6680 

6427 

6 
5 

56 

1247 

2.18 

7422 

3826 

4.21 

6174 

4 

57 

1378 

2.18 

7300 

4079 

4.21 

59" 

3 

58 

1509 

2.18 

7178 

4331 

4.21 

5669 

2 

59 

1640 

2.18 

7056 

2.03 

4584 

4.21 

5416 

1 

60 

9.841771 

9.856934 

9.984837 

10.015163 

0  : 

Cosine.  1  Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  I" 

Tang. 

M.  i 

1  133°                                       46°  1 

27 


85 


44°            I,OCb 

135° 

fLRITHIMEIC 

M. 

Sine. 

Diff.  V 

Co.-iiue. 

Diff.l' 

Tang. 

Diff.  1" 

Cotang. 

0 

9.841771 

2.l8 

9.856934 

2.03 

9.984837 

4.21 

10.015163 

60 

1 

1902 

2.18 

6812 

2.0C 

5090 

4.21 

4910 

59 

2 

2033 

2.18 

6690 

2.04 

5343 

4.21 

4657 

58 

3 

2163 

2.17 

6568 

5596 

4.21 

4404 

57 

4 

2294 

2.17 

6446 

5848 

4.21 

4152 

56 

5 

2424 

2.17 

6323 

6101 

4.21 

3899 

55 

6 

2^5 

2.17 

6201 

6354 

4.21 

3646 

54 

7 

2685 

2.17 

6078 

6607 

4.21 

3393 

53 

8 

2815 

2.17 

5956 

6860 

4.21 

3140 

52 

9 

2946 

2.17 

5833 

2.04 

7112 

4.21 

2888 

51 

10 

3076 

2.17 

5711 

3-05 

7365 

4.21 

2635 

50 

11 

9.843206 

2.16 

9.855588 

9.987618 

4.21 

10.012382 

49 

12 

3336 

2.16 

5465 

7871 

4.21 

2129 

48 

13 

3466 

2.16 

534^ 

8123 

4.21 

1877 

47 

14 

3595 

2.16 

5219 

8376 

4.21 

1624 

46 

15 

37^5 

2.16 

5096 

8629 

4.21 

I37i 

45 

16 

3855 

2.l6 

4973 

8882 

4.21 

1118 

44 

17 

3984 

2.16 

4850 

9J34 

4.21 

0866 

43 

18 

4114 

2.16 

4727 

2.05 

9387 

4.21 

0613 

42 

19 

4243 

2.I5 

4603 

2.06 

9640 

4.21 

0360 

41 

20 

4372 

2.15 

4480 

9.989893 

4.21 

10.010107 

40 

21 

9.844502 

2.15 

9.854356 

9.990145 

4.21 

10.009855 

39 

22 

4631 

2.15 

4233 

039» 

4.21 

9602 

38 

23 

4760 

2.15 

4109 

0651 

4.21 

9349 

37 

24 

4889 

2.15 

3986 

0903 

4.21 

9097 

36 

25 

5018 

2.15 

3862 

1156 

4.21 

8844 

35 

26 

5H7 

2.15 

3738 

2.06 

1409 

4.21 

8591 

34 

27 

5276 

2.14 

3614 

2.07 

1662 

4.21 

8338 

33 

28 

5405 

2.14 

349° 

1914 

4.21 

8086 

32 

29 

5533 

2.14 

3366 

2167 

4.21 

7833 

31 

30 

5662 

2.14 

3242 

2420 

4.21 

7580 

30 

~3l 

9.845790 

2.14 

9.853118 

9.992672 

4.21 

10.007328 

29 

32 

59i9 

2.14 

2994 

2925 

4.21 

7075 

28 

33 

6047 

2.14 

2869 

3178 

4.21 

6822 

27 

34 
35 

6175 
6304 

2.14 
2.14 

^745 
2620 

2.07 

343° 
3683 

4.21 
4.21 

6570 
6317 

26 
25 

36 

6432 

2.13 

2496 

2.08 

3936 

4.21 

6064 

24 

37 

6560 

2.13 

2371 

4189 

4.21 

5811 

23 

38 

6688 

2.13 

2247 

4441 

4.21 

5559 

22 

39 

6816 

2.13 

2122 

4694 

4.21 

5306 

21 

40 

6944 

2.13 

1997 

4947 

4.21 

5053 

20 

41 

9.847071 

2.13 

9.851872 

9.995199 

4.21 

10.004801 

19 

42 

7199 

2.13 

1747 

5452 

4.21 

4548 

18 

43 

73*7 

2.13 

l622 

2.08 

57°5 

4.21 

4295 

17 

44 

7454 

2.12 

M97 

2.09 

5957 

4.21 

4043 

16 

45 

75^ 

2.12 

1372 

6210 

4.21 

379° 

15 

46 

7709 

2.12 

1246 

6463 

4.21 

3537 

14 

47 

7836 

2.12 

II2I 

6715 

4.21 

3285 

13 

48 

7964 

2.12 

0996 

6968 

4.21 

3032 

12 

49 

8091 

2.12 

0870 

7221 

4.21 

2779 

11 

50 

8218 

2.12 

0745 

7473 

4.21 

2527 

10 

51 

9.848345 

2.12 

9.850619 

2.09 

9.997726 

4.21 

10.002274 

9 

52 

8472 

2.1  I 

0493 

2.10 

7979 

4.21 

2021 

8 

53 

8599 

2.II 

0368 

8231 

4.21 

1769 

7 

54 

8726 

2.II 

0242 

8484 

4.21 

I5l6 

6 

55 

8852 

2.1  I 

9.850116 

8737 

4.21 

1263 

5 

56 

8979 

2.  1  I 

9.849990 

8989 

4.21 

IOII 

4 

57 

9106 

2.  1  I 

9864 

9242 

4.21 

0758 

3 

58 

9232 

2.  1  1 

9738 

9495 

4.21 

0505 

2 

59 

9359 

2.  1  1 

9611 

2.10 

9.999747 

4.21 

0253 

1  j 

60 

9.849485 

9.849485 

IO.OOOOOO 

10.000000 

0 

Cosine. 

Diff.  1" 

Sine. 

Diff.l" 

Cotang. 

Diff.  1" 

Tang. 

M. 

134°                                       45° 

'TABLE 


OF 


NATURAL    SINES 


COSINES. 


«7 


NATURAL  SINES  AND  COSINES. 

f 

0° 

1° 

2° 

3° 

4° 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

•  Sine. 

Cosine. 

Sine. 

Cosine. 

~b~ 

00000 

Unit. 

01745 

99985 

03490 

99939 

05*34 

99863 

06976 

99756 

60 

i 

00029 

Unit. 

01774 

99984 

035*9 

99938 

05263 

99861 

07005 

99754 

59 

2 

00058 

Unit. 

01803 

99984 

03548 

99937 

05292 

99860 

07034 

99752 

58 

3 

00087 

Unit. 

01832 

99983 

03577 

99936 

05321 

99858 

07063 

99750 

57 

4 

00116 

Unit 

01862 

99983 

03606 

99935 

05350 

99857 

07092 

99748 

56 

5 

00145 

Unit. 

01891 

99982 

03635 

99934 

05379 

99855 

07121 

99746 

55 

6 

00175 

Unit. 

01920 

99982 

03664 

99933 

05408 

99854 

07150 

99744 

54 

7 

00204 

Unit. 

01949 

99981 

03693 

99932 

05437 

99852 

07179 

99742 

53 

8 

00233 

Unit. 

01978 

99980 

03723 

99931 

05466 

07208 

99740 

52 

9 

00262 

Unit. 

02007 

99980 

03752 

99930 

05495 

99849 

07237 

99738 

51 

10 

00291 

Unit. 

02036 

99979 

03781 

99929 

05524 

99847 

07266 

99736 

50 

11 

00320 

99999 

02065 

99979 

03810 

99927 

05553 

99846 

07295 

99734 

~49~ 

12 

00349 

99999 

02094 

99978 

03839 

99926 

05582 

99844 

07324 

99731 

48 

13 

00378 

99999 

02123 

99977 

03868 

999*5 

05611 

99842 

07353 

99729 

47 

14 
15 

00407 
00436 

99999 
99999 

02152 
02181 

99977 
99976 

03897 
03926 

99924 
99923 

05640 
05669 

99841 
99839 

07382 
07411 

99727 
99725 

46 
45 

16 

00465 

99999 

022X1 

99976 

03955 

99922 

05698 

99838 

07440 

99723 

44 

17 

00495 

99999 

02240 

99975 

03984 

99921 

05727 

99836 

07469 

99721 

43 

18 

00524 

99999 

02269 

99974 

04013 

99919 

05756 

99834 

07498 

99719 

42 

19 

00553 

99998 

02298 

99974 

04042 

999x8 

05785 

99833 

07527 

99716 

41 

20 

00582 

99998 

02327 

99973 

04071 

99917 

05814 

99831 

07556 

997*4 

40 

21 

00611 

99998 

02356 

99972 

04100 

999x6 

05844 

99829 

07585 

99712 

~39~ 

22 
23 

00640 
00669 

99998 
99998 

02385 
024X4 

99972 
99971 

04129 
04159 

99915 
999*3 

05873 
05902 

99827 
99826 

07614 
07643 

997*0 
99708 

38 
37 

24 

00698 

99998 

02443 

99970 

04188 

99912 

05931 

99824 

07672 

99705 

36 

25 

00727 

99997 

02472 

99969 

04217 

99911 

05960 

9982* 

07701 

99703 

35 

26 

00756 

99997 

O25OI 

99969 

04246 

99910 

05989 

99821 

07730 

99701 

34 

27 

00785 

99997 

02530 

99968 

04275 

99909 

06018 

99819 

07759 

99699 

33 

28 
29 
30 

00814 
00844 
00873 

99997 
99996 
99996 

02560 
02589 
026l8 

99967 
99966 
99966 

04304 

°4333 
04362 

999°7 
99906 
99905 

06047 
06076 
06105 

99817 
99815 
99813 

07788 
07817 
07846 

99696 
99694 
99692 

32 
31 
30 

31 

00902 

99996 

02647 

99965 

04391 

99904 

06134 

99812 

07875 

99689 

29 

32 

00931 

99996 

02676 

99964 

04420 

99902 

06163 

99810 

07904 

99687 

28 

33 

00960 

99995 

02705 

99963 

04449 

99901 

06192 

99808 

07933 

99685 

27 

34 

00989 

99995 

02734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

26 

35 

01018 

99995 

02763 

99962 

04507 

99898 

06250 

99804 

07991 

99680 

25 

36 

01047 

99995 

02792 

99961 

04536 

99897 

06279 

99803 

08020 

99678 

24 

37 

01076 

99994 

02821 

99960 

04565 

99896 

06308 

99801 

08049 

99676 

23 

38 

01105 

99994 

02850 

99959 

04594 

99894 

06337 

99799 

08078 

99673 

22 

39 

01134 

99994 

02879 

99959 

04623 

99893 

06366 

99797 

08107 

99671 

21 

40 

01164 

99993 

02908 

99958 

04653 

99892 

06395 

99795 

08136 

99668 

20 

41 

01193 

99993 

02938 

99957 

04682 

9989° 

06424 

99793 

08165 

99666 

19 

42 

01222 

99993 

02967 

99956 

04711 

99889 

06453 

99792 

08194 

99664 

18 

43 

OI25I 

99992 

02996 

99955 

04740 

99888 

06482 

99790 

08223 

99661 

17 

44 

01280 

99992 

03025 

99954 

04769 

99886 

06511 

99788 

08252 

99659 

16 

45 

01309 

99991 

03054 

99953 

04798 

99885 

06540 

99786 

08281 

99657 

15 

46 

01338 

99991 

03083 

99952 

04827 

99883 

06569 

99784 

08310 

99654 

14 

47 

01367 

99991 

03II2 

99952 

04856 

99882 

06598 

99782 

08339 

99652 

13 

48 

01396 

99990 

03X41 

99951 

04885 

99881 

06627 

99780 

08368 

99649 

12 

49 

01425 

99990 

03170 

99950 

04914 

99879 

06656 

99778 

08397 

99647 

11 

50 

01454 

99989 

03X99 

99949 

04943 

99878 

06685 

99776 

08426 

99_6_44 

10 

51 

01483 

99989 

03228 

99948 

04972 

99876 

06714 

99774 

08455 

99642 

9 

52 

OI5I3 

99989 

03257 

99947 

05001 

99875 

06743 

99772 

08484 

99639 

8 

53 

01542 

99988 

03286 

99946 

05030 

99873 

06773 

99770 

08513 

99637 

7 

54 

OI57I 

99988 

03316 

99945 

05059 

99872 

06802 

99768 

08542 

99635 

6 

55 

Ol6oO 

99987 

03345 

99944 

05088 

99870 

06831 

99766 

08571 

99632 

5 

56 

01629 

99987 

03374 

99943 

05117 

99869 

06860 

99764 

08600 

99630 

4 

1  57 

01658 

99986 

03403 

99942 

05146 

99867 

06889 

99762 

08629 

99627 

3 

58 

01687 

99986 

03432 

99941 

05175 

99866 

06918 

99760 

08658 

99625 

2 

59 

01716 

99985 

03461 

99940 

05205 

99864 

06947 

99758 

08687 

99622 

1 

60 

01745 

99985 

03490 

99939 

05234 

99863 

06976 

99756 

08716 

99619 

0 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

89° 

88° 

87° 

86° 

85° 

NATURAL  SINES  AND  COSINES. 

/ 

5° 

6° 

7° 

8° 

9° 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

e 

08716 

99619 

*°453 

99452! 

12187 

99255 

*39*7 

99027 

*5643 

98769 

60 

i 

08745 

99617 

10482 

99449 

12216 

99251 

13946 

99023 

15672 

98764 

59 

2 

08774 

99614 

10511 

99446 

12245 

99248 

*3975 

990*9 

15701 

98760 

58 

3 

08803 

99612 

10540 

99443 

12274 

99244 

14004 

990*5 

15730 

98755 

57 

4 

08831 

99609 

10569 

99440 

12302 

99240 

14033 

99011 

15758 

98751 

56 

5 

08860 

99607 

10597 

99437 

12331 

99237 

14061 

99006 

15787 

98746 

55 

6 

08889 

99604 

10626 

99434 

12360 

99233 

14090 

99002 

15816 

98741 

54 

7 

08918 

99602 

10655 

9943* 

12389 

99230 

*4**9 

98998 

15845 

98737 

53 

8 

08947 

99599 

10684 

99428 

12418 

99226 

14148 

98994 

15873 

98732 

52 

9 

08976 

99596 

107*3 

99424 

12447 

99222 

14177 

98990 

15902 

98728 

51 

10 

09005 

99594 

10742 

99421 

12476 

99219 

14205 

98986 

1593* 

98723 

50 

11 

09034  9959* 

10771 

994*8 

12504 

992*5 

14234 

98982 

15959 

98718 

49 

12 

09063(99588 

10800 

994*5 

12533 

99211 

14263 

98978 

15988 

987*4 

48 

13 

09092(99586 

10829 

994*2 

12562 

99208 

14292 

98973 

16017 

98709 

47 

14 

09121 

99583 

10858 

99409 

12591 

99204 

14320 

98969 

16046 

98704 

46 

15 

09150 

99580 

10887 

99406 

12620 

99200 

*4349 

98965 

16074 

98700 

45 

16 

09179 

99578 

10916 

99402 

12649 

99*97 

14378 

98961 

16103 

98695 

44 

17 

09208 

99575 

10945 

99399 

12678 

99*93 

14407 

98957 

16132 

98690 

43 

18 

09237 

99572 

*°973 

99396 

12706 

99*89 

14436 

98953 

16160 

98686 

42 

19 

09266 

99570 

1  1  002 

99393 

12735 

99186 

14464 

98948 

16189 

98681 

41 

20 

09295 

99567 

II03I 

99390 

12764 

99182 

*4493 

98944 

16218 

98676 

40 

21 

09324 

99564 

11060 

99386 

12793 

99*78 

14522 

98940 

16246 

98671 

39 

22 

09353 

99562 

11089 

99383 

12822 

99*75 

*455* 

98936 

16275 

98667 

38 

23 

09382 

99559 

11118 

99380 

12851 

99171 

14580 

98931 

16304 

98662 

37 

24 

09411 

99556 

ii*47 

99377 

12880 

99*67 

14608 

98927 

98657 

36 

25 

09440 

99553 

11176 

99374 

12908 

99*63 

14637 

98923 

16361 

98652 

35 

26 

09469 

9955* 

11205 

99370 

12937 

99160 

14666 

98919 

16390 

98648 

34 

27 

09498 

99548 

11234 

99367 

12966 

99*56 

14695 

989*4 

16419 

98643 

33 

28 

09527 

99545 

11263 

99364 

12995 

99*52 

*4723 

98910 

16447 

98638 

32 

29 

09556 

99542 

11291 

99360 

13024 

99*48 

14752 

98906 

16476 

98633 

31 

30 

09585 

99540 

11320 

99357 

13053 

99*44 

14781 

98902 

16505 

98629 

30 

~B1 

09614 

99537 

1*349 

99354 

13081 

99141 

14810 

98897 

16533 

98624 

29 

32 

09642 

99534 

H378 

9935* 

13110 

99*37 

14838 

98893 

16562 

98619 

28 

33 

09671 

9953* 

11407 

99347 

13139 

99*33 

14867 

98889 

16591 

98614 

27 

34 

09700 

99528 

11436 

99344 

13168 

99*29 

14896 

98884 

16620 

98609 

26 

35 

09729 

99526 

11465 

9934* 

13197 

99*25 

14925 

98880 

16648 

98604 

25 

36 

09758 

99523 

i*494 

99337 

13226 

99122 

*4954 

98876 

16677 

98600 

24 

37 

09787 

99520 

11523 

99334 

13254 

99118 

14982 

98871 

16706 

98595 

23 

38 

09816 

995*7 

i*552 

9933* 

13283 

99114 

15011 

98867 

16734 

98590 

22 

39 

09845 

995*4 

11580 

99327 

13312 

991  10 

15040 

98863 

16763 

98585 

21 

40 

09874 

995** 

11609 

99324 

13341 

99106 

15069 

98858 

16792 

98580 

20 

41 

09903 

99508 

11638 

99320 

13370 

99102 

15097 

98854 

16820 

98575 

19 

42 

09932 

99506 

1  1667 

993*7 

*3399 

99098 

15126 

98849 

16849 

98570 

18 

43 

09961 

99503 

11696 

993*4 

*3427 

99094 

*5*55 

98845 

16878 

98565 

17 

44 

09990 

99500 

1*725 

993*0 

13456 

99091 

15184 

98841 

16906 

98561 

16 

45 

10019 

99497 

**754 

993°7 

*3485 

99087 

15212 

98836 

16935 

98556 

15 

46 

10048 

99494 

11783 

99303 

13514 

99083 

15241 

98832 

16964 

98551 

14 

47 

10077 

99491 

11812 

99300 

*3543 

99°79 

15270 

98827 

16992 

98546 

13 

48 

10106 

99488 

11840 

99297 

13572 

99075 

15299 

98823 

17021 

98541 

12 

49 

10135 

99485 

11869 

99293 

13600 

99071 

*5327 

98818 

17050 

98536 

11 

50 

10164 

99482 

11898 

99290 

13629 

99067 

15356 

98814 

17078 

98531 

10 

51 

10192 

99479 

11927 

99286 

13658 

99063 

*5385 

98809 

17107 

98526 

9 

52 

I022I 

99476 

11956 

99283 

13687 

99059 

15414 

98805 

17136 

98521 

8 

53 

IO25O 

99473 

11985 

99279 

13716 

99055 

15442 

98800 

17164 

98516 

7 

54 

10279 

99470 

12014 

99276 

*3744 

99051 

15471 

98796 

17193 

98511 

6 

55 

10308 

99467 

12043 

99272 

*3773 

99047 

15500 

98791 

17222 

98506 

5 

56 

10337 

99464 

12071 

99269 

13802 

99043 

15529 

98787 

17250 

98501 

4 

1  57 

10366 

99461 

I2IOO 

99265 

1383* 

99039 

*5557 

98782 

17279 

98496 

3 

i  58 

10395 

99458 

I2I29 

99262 

13860 

99035 

15586 

98778 

17308 

98491 

2 

59 

10424 

99455 

I2I58 

99258 

13889 

99031 

15615 

98773 

17336 

98486 

1 

60 

10453 

99452 

I2I87 

99255 

99027 

*5643 

98769 

17365 

98481 

0 

f 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

84° 

83° 

82° 

81° 

80° 

NATURAL  SIBXES  AKD  CO3IBJES. 

10° 

11° 

12° 

13° 

14° 

f 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

o 

7365 

98481 

19081 

98163 

20791 

97815 

22495 

97437 

24192 

97030 

60 

i 

7393 

98476 

19109 

98157 

20820 

97809 

22523 

97430 

24220 

97023 

59 

2 

7422 

98471 

19138 

98152 

20848 

97803 

22552 

97424 

24249 

97015 

58 

3 

745* 

98466 

19167 

98146 

20877 

97797 

22580 

974*7 

24277 

97008 

57 

4 

7479 

98461 

*9*95 

98140 

20905 

9779* 

22608 

974** 

24305 

97001 

56 

5 

7508 

98455 

19224 

98135 

20933 

97784 

22637 

97404 

24333 

96994 

55 

6 

7537 

98450 

19252 

98129 

20962 

97778 

22665 

97398 

24362 

96987 

54 

7 

7565 

98445 

19281 

98124 

20990 

97772 

22693 

97391 

24390 

96980 

53 

8 

7594 

98440 

19309 

98118 

21019 

97766 

22722 

97384 

24418 

96973 

52 

9 

7623 

98435 

19338 

98112 

21047 

97760 

22750 

97378 

24446 

96966 

51 

10 

7651 

98430 

19366 

98107 

21076 

97754 

22778 

9737* 

24474 

96959 

50 

11 

7680 

98425 

*9395 

98101 

21104 

97748 

22807 

97365 

24503 

96952 

49 

12 

17708 

98420 

19423 

98096 

21132 

97742 

22835 

97358 

2453* 

96945 

48 

13 

*7737 

98414 

19452 

98090 

2Il6l 

97735 

22863 

9735* 

24559 

96937 

47 

14 

17766 

98409 

19481 

98084 

21189 

97729 

22892 

97345 

24587 

96930 

46 

15 

*7794 

98404 

19509 

98079 

2I2I8 

97723 

22920 

97338 

24615 

96923 

45 

16 

17823 

98399 

19538 

98073 

21246 

977*7 

22948 

9733* 

24644 

96916 

44 

17 

17852 

98394 

19566 

98067 

21275 

977** 

22977 

97325 

24672 

96909 

43 

18 

17880 

98389 

*9595 

98061 

21303 

97705 

23005 

973*8 

24700 

96902 

42 

19 

17909 

9838, 

19623 

98056 

21331 

97698 

23033 

97311 

24728 

96894 

41 

20 

*7937 

98378 

19652 

98050 

21360 

97692 

23062 

97304 

24756 

96887 

40 

21 

17966 

98373 

19680 

98044 

21388 

97686 

23090 

97298 

24784 

96880 

~39~ 

22 

*7995 

98368 

19709 

98039 

2*4*7 

97680 

23118 

97291 

24813 

96873 

38 

23 

18023 

98362 

19737 

98033 

2*445 

97673 

23146 

97284 

24841 

96866 

37 

24 

18052 

98357 

19766 

98027 

2*474 

97667 

23*75 

97278 

24869 

96858 

36 

25 

18081 

98352 

*9794 

98021 

21502 

97661 

23203 

97271 

24897 

96851 

35 

26 

18109 

98347 

19823 

98016 

21530 

97655 

23231 

97264 

24925 

96844 

34 

27 

18138 

98341 

19851 

98010 

2*559 

97648 

23260 

97257 

24954 

96837 

33 

28 

18166 

98336 

19880 

98004 

2*587 

97642 

23288 

9725* 

24982 

96829 

32 

29 

18195 

98331 

19908 

97998 

21616 

97636 

23316 

97244 

25010 

96822 

31 

30 

18224 

98325 

*9937 

97992 

21644 

97630 

23345 

97237 

25038 

96815 

30 

31 

18252 

98320 

19965 

97987 

21672 

97623 

23373 

97230 

25066 

96807 

29 

32 

18281 

983*5- 

*9994 

97981 

21701 

97617 

23401 

97223 

25094 

96800 

28 

33 

18309 

98310 

2OO22 

97975 

21729 

97611 

23429 

97217 

25122 

96793 

27 

34 
35 

18338 
18367 

98304 
98299 

20051 
20079 

97969 
97963 

21758 
21786 

97604 
97598 

23458 
23486 

97210 
97203 

25151 
25*79 

9678^ 
96778 

26 
25 

36 

i8395 

98294 

20108 

97958 

21814 

97592 

235H 

97196 

25207 

96771 

24 

37 

18424 

98288 

20136 

97952 

21843 

97585 

23542 

97*89 

25235 

96764 

23 

38 

18452 

98283 

20165 

97946 

21871 

97579 

23571 

97182 

25263 

96756 

22 

39 

18481 

98277 

20193 

97940 

21899 

97573 

23599 

97*76 

25291 

96749 

21 

40 

18509 

98*72 

2O222 

97934 

21928 

97566 

23627 

97*69 

25320 

96742 

20 

41 

18538 

98267 

20250 

97928 

21956 

97560 

23656 

97162 

25348 

96734 

19 

42 

18567 

98261 

20279 

97922 

21985 

97553 

23684 

97*55 

25376 

96727 

18 

43 

i8595 

98256 

20307 

979*6 

22013 

97547 

23712 

97*48 

25404 

96719 

17 

44 

18624 

98250 

20336 

979*0 

22041 

9754* 

23740 

97141 

25432 

96712 

16 

45 

18652 

98245 

20364 

97905 

22070 

97534 

23769 

97*34 

25460 

96705 

15 

46 

18681 

98240 

20393 

97899 

22098 

97528 

23797 

97*27 

25488 

96697 

14 

47 

18710 

98234 

20421 

97893 

22126 

97521 

23825 

97120 

255*6 

96690 

13 

48 

18738 

98229 

20450 

97887 

22155 

975*5 

23853 

97**3 

25545 

96682 

12 

49 

18767 

98223 

20478 

97881 

22183 

97508 

23882 

97106 

25573 

96675 

11 

60 

i8795 

98218 

20507 

97875 

22212 

97502 

239*0 

97100 

25601 

96667 

10 

51 

18824 

98212 

20535 

97869 

2224O 

97496 

23938 

97093 

25629 

96660 

gT 

52 

18852 

98207 

2056^ 

97863 

22268 

97489 

23966 

97086 

25657 

96653 

8 

53 

18881 

98201 

20592 

97857 

22297 

97483 

23995 

97079 

25685 

96645 

7 

54 

18910 

98196 

20620 

97851 

22325 

97476 

2402-: 

97072 

257*3 

96638 

6 

55 

18938 

98190 

20649 

97845 

22353 

97470 

24051 

97065 

25741 

96630 

5 

56 

18967 

98185 

20677 

97839 

22382 

97463 

24079 

97058 

25769 

96623 

4 

57 

18995 

98179 

20706 

97833 

22410 

97457 

24108 

97051 

25798 

96615 

8 

58 

19024 

98174 

20734 

97827 

22438 

9745°  i  24136 

97044 

25826  96608 

2 

59 

19052 

98168 

[20763 

97821 

22467 

97444!  24164 

97037 

25854 

96600 

1 

60 

19081 

98*63 

20791 

978*5 

22495 

97437 

24192 

97030 

25882 

96593 

0 

Cosine 

Sine. 

Cosine 

Sine. 

Cosine. 

Sine. 

Cosine 

Sine. 

Cosine. 

Sine. 

79° 

78° 

77° 

76° 

75° 

90 


NATURAL  SINES  AND  COSINES. 

15° 

16° 

17° 

18° 

19° 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

0 

25882 

96593 

27564 

96126 

29237 

95630 

30902 

95106 

32557 

94552 

60 

i 

25910 

96585 

27592 

96118 

29265 

95622 

30929 

95097 

32584 

94542 

59 

2 

25938 

96578 

27620 

96110 

29293 

95613 

3°957 

95088 

32612 

94533 

58 

3 

25966 

96570 

27648 

96102 

29321 

95605 

30985 

95079 

32639 

94523 

57 

4 

25994 

96562 

27676 

96094 

29348 

95596 

31012 

95070 

32667 

945*4 

56 

5 

26022 

96555 

27704 

96086 

29376 

95588 

31040 

95061 

32694 

94504 

55 

6 

26050 

96547 

27731 

96078 

29404 

95579 

31068 

95052 

32722 

94495 

54 

V 

26079196540 

27759 

96070 

29432 

9557* 

3*095 

95043 

32749 

94485 

53 

8 

26107  96532 

27787 

96062 

29460 

95562 

31123 

950331 

32777 

94476 

52 

9 
10 

26135  96524 
26163(96517 

27815 
27843 

& 

29487 
29515 

95554 
95545 

31151 

31178 

95024 
950*5 

32804 
32832 

94466 
94457 

51 
50 

11 

26191  96509 

27871 

96037 

29543 

95536 

31206 

95006 

32859 

94447 

49  1 

12 

26219  96502 

27899 

96029 

29571 

95528 

3*233 

94997 

32887 

94438 

48  j 

13 

26247196494 

27927 

96021 

29599 

955*9 

31261 

94988 

329*4 

94428 

47 

14 

26275  96486 

27955 

96013 

29626 

955** 

31289 

94979 

32942 

944*8 

46 

15 

26303  96479 

27983 

96005 

29654 

955°2 

3*3*6 

94970 

32969 

94409 

45 

16 

26331  96471 

28011 

95997 

29682 

95493 

3*344 

94961 

32997  94399 

44 

17 
18 

26359  96463 
26387  96456 

28039 
28067 

95989 
95981 

29710 
29737 

95485 
95476 

3*372 
3*399 

94952 
94943 

33024 
33051 

94390 
94380 

43  i 
42 

19 
20 

26415,96448 
26443196440 

28095 
28123 

95972 
95964 

29765 
29793 

95467 
95459 

3*427 
3*454 

94933 
94924 

33079 
33106 

94370 
94361 

41 

40 

21 

22 

26471 
26500 

96433 
96425 

28150 

28178 

95956 
95948 

29821 
29849 

9545° 
9544* 

31482 
3*5*o 

949*5 
94906 

33134 
33161 

9435* 
94342 

39 

38 

23 

26528 

964*7 

28206 

95940 

29876 

95433 

3*537 

94897 

33*89 

94332 

37 

24 

26556 

96410 

28234 

9593* 

29904 

95424 

3*565 

94888 

33216 

94322 

36 

25 

26584 

96402 

28262 

95923 

29932 

954*5 

3*593 

94878 

33244 

943*3 

35 

26 

26612 

96394 

28290 

959*5 

29960 

95407 

31620 

94869 

33271 

94303 

34 

27 

26640 

96386 

28318 

95907 

129987 

95398 

31648 

94860 

33298 

94293 

33 

28 

26668 

96379 

28346 

95898 

130015 

95389 

3*675 

94851 

33326 

94284 

32 

29 

26696 

96371 

28374 

95890 

30043 

95380 

31703 

94842 

33353 

94274 

31 

1  30 

26724 

96363 

28402 

95882 

30071 

95372 

3*73° 

94832 

33381 

94264 

30 

31 

26752 

96355 

28429 

95874 

30098 

95363 

31758 

94823 

334°8 

94254 

29 

32 

26780 

96347 

28457 

95865 

30126 

95354 

31786 

94814 

33436 

94245 

28  1 

33 

26808 

96340 

28485 

95857 

30154 

95345 

31813 

94805 

3346T 

74235 

27  j 

34 

26836 

96332 

285*3 

95849 

30182 

95337 

31841 

94795 

33490 

94225 

26 

35 

26864 

96324 

28541 

95841 

30209 

95328 

31868 

94786 

335*8 

942*5 

25 

36 

26892 

96316 

28569 

95832 

30237 

953*9 

31896 

94777 

33545 

94206 

24 

37 

26920 

96308 

28597 

95824 

30265 

953*0 

31923 

94768 

33573 

94*96 

23 

38 

26948 

96301 

28625 

95816 

30292 

95301 

3*95* 

94758 

33600 

94*86 

22 

39 

26976 

96293 

28652 

95807 

30320 

95293 

3*979 

94749 

33627 

94176 

21 

1  40 

27004 

96285 

28680 

95799 

30348 

95284 

32006 

94740 

33655 

94167 

20 

41 

27032 

96277 

28708 

9579* 

30376 

95275 

32034 

94730 

33682 

94*57 

19 

42 

27060 

96269 

28736 

95782 

30403 

95266  jj  32061 

94721 

337*0 

94*47 

18 

43 

27088 

96261 

28764 

95774 

30431 

95257 

32089 

947*2 

33737 

94*37 

17 

44 

27116 

96253 

28792 

95766 

30459 

95248 

321  16 

94702 

33764 

94*27 

16 

45 

27144 

96246 

28820 

95757 

30486 

95240 

32*44 

94693 

33792 

94118 

15 

46 

27172 

96238 

28847 

95749 

30514 

95231  32171 

94684 

338*9 

94108 

14 

47 
48 

27200 
27228 

96230 
96222 

28875 
28903 

9574° 
95732 

30542  95222  32199 
30570  952131132227 

94674 
94665 

33846 

33874 

94098 
94088 

13  1 
12 

49  '127256  96214 
50  127284  96206 

28931 
28959 

95724 
957*5 

30597 
30625 

95204  32254 
95195  132282 

94656 
94646 

33901 
33929 

94078 
94068 

11  | 
10 

51  27312  96198 

28987 

95707 

30653 

95186)132309 

94637 

33956 

94058 

9 

52 
53 
54 
55 

27340196190 
27368  96182 
27396  96174 
27424  96166 

29015 

29042 
29070 
29098 

95698 
95690 
95681 
95673 

30680 
30708 
30736 
30763 

95*77 
95168 

95*59 
95150 

32337 
32364 
32392 
1324*9 

94627  33983 
94618  34011 
(94609  34038 
J94599  34o65 

94049 
94039 
94029 
940*9 

8 
7 
6 
5 

56 

27452  96158 

29126 

95664 

30791 

95142  |  32447 

94590  i  34093 

94009 

4 

57 

27480  96150 

29*54 

95656 

30819 

95*33 

32474 

94580  134120 

93999 

3 

58 

27508  96142 

29182 

95647 

30846  95124 

32502194571  134147 

93989 

2  j 

59 
60 

27536  96134 
27564  96126 

29209 
29237 

95639 
9563° 

30874 
30902 

95**5 
95106 

32529(94561  134175 
32557  945521  34202 

93979 
93969 

1 
0 

Cosine.  Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine 

Cosine. 

Sine. 

1            I 

i 

74°       73° 

72° 

71° 

I   70° 

91 


NATURAL  SINES  AND  COSINES. 

20°       21° 

22°   ||   23° 

24° 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine.  'Cosine. 

,,„,. 

Cosine. 

Sine. 

Cosine. 

0 

34202 

93969 

35837 

93358 

3746i;927i8 

39073 

92050 

40674 

9*355 

60 

i 

34229  93959 

35864 

93348 

37488 

92707 

39*0° 

92039 

40700 

9*343 

59 

34257  93949 

35891 

93337 

375*5 

92697 

39*27 

92028 

40727 

9*33* 

58 

3 

34284  93939 

359*8 

93327 

37542 

92686 

39*53 

92016 

40753 

9*3*9 

57 

4  J343*i  93929 

35945 

933*6 

37569 

92675 

39180 

92005 

40780 

9*307 

56 

5 

34339 

939*9 

35973 

93306 

37595 

92664 

39207 

9*994 

40806 

9*295 

55 

6 

34366 

93909 

36000 

93295 

37622 

92653! 

39234 

91982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

9*97* 

40860 

91272 

53 

8 

3442* 

93889 

36054 

93274 

37676 

92631 

39287 

9*959 

40886 

91260 

52 

9 
10 

34448  93879 
34475  93869 

36081 
36108 

93264 
93253 

37703 
3773° 

92620 
92609 

393*4 
3934* 

9*948 
9*936 

40913 
40939 

91248 
9*236 

51 

50 

11 

345°3  93859 

36*35 

93243 

37757 

92598 

39367 

9*925 

40966 

91224 

49 

12 

3453° 

93849 

36162 

93232 

37784 

92587 

39394 

9*9*4 

40992 

91212 

48 

13 

34557 

93839 

36190 

93222 

37811 

92576 

39421 

91902 

41019 

91200 

47 

14 
15 

34584 
34612 

93829 
93819 

36217 
36244 

932** 
93201 

37838 
37865 

92565 
92554 

39448 
39474 

91891 
9*879 

4*045 
41072 

91188 
91176 

46 
45 

16 

34639 

93809 

36271 

93190 

37892 

92543 

395oi 

91868 

41098 

91164 

44 

17 

34666 

93799 

36298 

93180 

379*9 

92532 

39528 

91856 

4*125 

91152 

43 

18 
19 

34694  93789 
34721  93779 

36325 
36352 

93*69 
93*59 

37946 
37973 

92521 
92510 

39555 
3958i 

9*845 
9*833 

41151 

41*78 

91140 
91128 

42 
41 

20 

34748 

93769 

36379 

93*48 

37999 

92499 

39608 

91822 

41204 

91116 

40 

21 

34775 

93759 

36406 

93*37 

38026 

92488 

39635 

91810 

41231 

91  104 

39 

22 

34803 

93748 

36434 

93*27 

38053 

92477 

39661 

9*799 

4*257 

91092 

38 

23 

34830 

9373s 

36461 

93**6 

38080 

92466 

39688 

9*787 

41284 

91080 

37 

24 
25 

34857 
34884 

93728 
93718 

36488 
365*5 

93106 
93095 

38107 

92455 
92444 

397*5 
3974* 

9*775 
91764 

4*3*0 

4*337 

91068 
91056 

36 
35 

26 
27 

349*2 
34939 

93708 
93698 

36569 

93084 
93074 

38161 
38188 

92432 
92421 

39768 
39795 

9*752 
91741 

4*363 
41390 

91044 
91032 

34 
33 

28 

34966 

93688 

36596 

93063 

382*5 

92410 

39822 

9*729 

41416 

91020 

32 

29 

34993 

93677 

36623 

93052 

38241 

92399 

39848 

9*7*8 

4*443 

91008 

31 

30 

35021 

93667 

36650 

93042 

38268 

92388 

39875 

91706! 

4*469 

90996 

30 

31 

35048 

93657 

36677 

93°3* 

38295 

92377 

39902 

9*694 

4*496 

90984 

29 

32 

35°75 

93647 

36704 

93020 

38322 

92366 

39928 

9*683! 

41522 

90972 

28 

33 

35102 

93637 

36731 

93010 

38349 

92355 

39955 

91671 

4*549 

90960 

27 

34 

35*3° 

93626 

36758 

92999 

38376 

92343 

39982 

91660 

4*575 

90948 

26 

35 

35*57 

93616 

36785 

92988 

38403 

92332 

40008 

91648 

41602 

90936 

25 

36 

35*84 

93606 

36812 

92978 

38430 

92321 

40035 

91636 

41628 

90924 

24 

37 

352*1 

93596 

36839 

92967 

38456 

923*0 

40062 

91625 

4*655 

90911 

23 

38 

35239 

93585 

36867 

92956 

38483 

92299 

40088 

9*6*3 

41681 

90899 

22 

39 

35266 

93575 

36894 

92945 

385*0 

92287 

40115 

91601 

4*707 

90887 

21 

40 

35293 

93565 

36921 

92935 

38537 

92276 

40141 

91590 

4*734 

90875 

20  | 

41 

35320 

93555 

36948 

92924 

38564 

92265 

40168 

9*578 

41760 

90863 

19 

42 

35347 

93544 

36975 

929*3 

3859* 

92254 

40195 

91566 

41787 

90851 

18 

43 

35375 

93534 

37002 

92902 

38617 

92243 

40221 

9*555 

41813 

90839 

17 

44 

35402 

93524 

37029 

92892 

38644 

92231 

40248 

9*543 

41840 

90826 

16  1 

45 

35429 

935*4 

37056 

92881 

38671 

92220 

40275 

9*53* 

41866 

90814 

15 

46 

35456 

93503 

37083 

92870 

38698 

92209 

40301 

9*5*9 

41892 

90802 

14 

1  47 

35484 

93493 

37*10 

92859 

38725 

92198 

40328 

91508 

4*9*9 

90790 

13 

48 

3551* 

93483 

37*37 

92849 

38752 

92186 

40355 

9*496 

4*945 

90778 

12 

49 

35538 

93472 

37*64 

92838 

38778 

92*75 

40381 

91484 

4*972 

90766 

11 

50 

35565 

93462 

92827 

38805 

92164 

40408 

91472 

4*998 

90753 

10 

51 

35592 

93452 

37218 

92816 

38832 

92152 

40434 

91461 

42024 

90741 

9 

52 
53 

35619  93441 
3564719343* 

37245 
37272 

92805 
92794 

38886 

92141 
92130 

40461 
40488 

9*449 
9*437 

42051 
42077 

90729 
90717 

8 

7 

i  54 

35674193420 

37299 

92784 

389*2 

92119 

40514 

9*425 

42104 

90704 

6  1 

55 

3570*193410 

37326 

92773 

38939 

92107 

I4°54* 

9*4*4 

42130 

90692 

5 

1  56 

35728 

93400 

37353 

92762 

38966 

92096 

40567 

91402 

42156 

90680 

4 

57 

35755 

93389 

9275* 

38993 

92085 

40594 

9*390 

42183 

90668 

3 

1  58 

35782 

93379 

37407 

92740 

39020 

92073 

40621 

91378 

42209590655 

2 

59 

35810 

93368 

37434 

92729 

39046 

92062 

40647 

91366 

42235 

90643 

1  1 

60 

35837 

93358 

92718 

39073 

92050 

40674 

9*355 

42262 

90631 

0  1 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

69° 

68° 

67° 

66° 

65° 

92 


NATURAL  SIEVES  A1TO  COSINES. 

/ 

25° 

26° 

27° 

28° 

29° 

/ 

~w 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

0 

i 

2 
3 
4 
5 
6 
7 
8 
9 
10 

42262  90631 
42288  90618 
42315  90606 
42341190594 
42367^0582 
42394  90569 
42420190557 
42446  90545 
42473  90532 
42499:90520 
42525  90507 

43863 
43889 
43916 
43942 
43968 

43994 
44020 
44046 
44072 
44098 

89879 
89867 

898.54 
89841 
89828 
89816 
89803 
89790 
89777 
89764 
89752 

45399 

45425 
45451 

45477 
45503 
45529 

45554 
45580 
45606 

45632 
45658 

89101 
89087 
89074 
89061 
89048 
89035 
89021 
89008 
88995 
88981 
88968 

46947 
46973 
46999 
47024 
47050 
47076 
47101 
47127 

47153 
47178 
47204 

88295 
88281 
88267 
88254 
88240 
88226 

88213 
88199 
88185 
88172 
88158 

48481' 
48506 
48532 
48557 
48583 
48608 

48634 
48659 
48684 
48710 
48735 

87462 
87448 

87434 
87420 
87406 
87391 
87377 
87363 
87349 
87335 
87321 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

42552 
42578 
42604 

42657 
42683 
42709 
42736 
42762 
42788 

90495 
90483 
90470 
90458 
90446 

90433 
90421 
90408 
90396 
90383 

44124 

44151 

44177 
44203 
44229 

44255 
44281 

44307 
44333 
44359 

89739 
89726 
89713 
89700 
89687 

89674 
89662 

89649 
89636 
89623 

45684 

45710 

45736 
45762 

45787 
45813 
45839 
45865 
45891 

459*7 

88955 
88942 
88928 
88915 
88902 
88888 
88875 
88862 
88848 
88835 

47229 

47255 
47281 
47306 
47332 

47358 
47383 
47409 

47434 
47460 

88144 
88130 
88117 
88103 
88089 

88075 
88062 
88048 
88034 
88020 

48761 
48786 
48811 
48837 
48862 

48888 
48913 
48938 
48964 
48989 

49014 

49040 
49065 
49090 
49116 
49141 
49166 
49192 
49217 
49242 
49268 

49293 
49318 

49344 
49369 

49394 
49419 

49445 
49470 

49495 

87306 
87292 
87278 
87264 
87250 

87235 
87221 
87207 

87193 
87178 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 

22 
23 
24 
25 
26 
27 
28 
29 
30 

42815 
42841 
42867 
42894 
42920 
42946 
42972 
42999 
43025 
4305i 

90371 
90358 
90346 

9°334 
90321 

90309 
90296 
90284 
90271 
90259 

44385 
44411 

44437 
44464 
44490 

445  1  6 
44542 
44568 

44594 
44620 

89610 

89597 
89584 
89571 
89558 

89545 
89532 
89519 
89506 
89493 

45942 
45968 

45994 
46020 
46046 
46072 
46097 
46123 
46149 
46175 

88822 
88808 
88795 
88782 
88768 
88755 
88741 
88728 
88715 
88701 

47486 
47511 
47537 
47562 
47588 
47614 

47639 
47665 
47690 
47716 

88006 
87993 
87979 

87965 
87951 

87937 
87923 
87909 
87896 
87882 

87164 
87150 
87136 
87121 
87107 

87093 
87079 
87064 
87050 
87036 

87021 
87007 
86993 
86978 
86964 
86949 
86935 
86921 
86906 
86892 
86878 
86863 
86849 
86834 
86820 
86805 
86791 
86777 
86762 
86748 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

i  31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

43°77 
43I04 
43130 
43156 
43182 
43209 

43235 
43261 
43287 
43313 

4334° 
43366 

4339^ 
434i8 
43445 
43471 
43497 
43523 
43549 
43575 

90246 
90233 
90221 
90208 
90196 
90183 
90171 
90158 
90146 
90*33 
90120 
90108 
90095 
90082 
90070: 
90057 

90045  ! 
90032  i 
9OOI9i 
90007  | 

44646 

44672 
44698 
44724 
4475° 

44776 
44802 
44828 

44854 
44880 

89480 
89467 
89454 
89441 
89428 
89415 
89402 
89389 
89376 
89363 

46201 
46226 
46252 
46278 
46304 

4633° 
46355 
46381 
46407 

46433 

88688 
88674 
88661 
88647 
88634 

88620 
88607 

88593 
88580 
88566 

47741 
47767 

47793 
47818 

47844 
47869 

47895 
47920 

47946 
47971 

87868; 
87854 
87840 
87826 
87812 

87798 
87784! 
87770 
87756| 
87743 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

61 

52 
53 
54 
55 
56 
57 
58 
'  59 
60 

/ 

44906 
44932 
44958 

44984 
45010 

45036 
45062 
45088 
45114 
45140 

89350 
89337 
89324 
89311 
89298 
89285 
89272 
89259 
89245 
89232 

46458 
46484 
46510 
46536 
46561 

46587 
46613 
46639 
46664 
46690 

88553 
88539 
88526 
88512 
88499 
88485 
88472 
88458 

88445 
88431 

47997 
48022 
48048 
48073 
48099 

48124 
48150 
48175 
48201 
48226 

877291 
87715 
87701 
87687 
87673 

87659! 

87645! 
87631  | 
87617! 
87603 

49521 
49546 
49571 
49596 
49622 

49647 
49672 
49697 

49723 
49748 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 
7 
6 
5 
4 
3 
2 
I 
0 

/ 

43602  89994 
43628189981  1 
43654  89968 
43680  89956 
43706  89943 

43733  89930 
43759i899l8 
43785  89905 
43811189892 
43837  89879 

45166 
45192 
45218 

45243 
45269 

45295 
45321 
45347 
45373 
45399 

89219 
89206 
89193 
89180 
89167 

89153 
89140 
89127 
89114 
89101 

46716 

46767 
46793 
46819 
46844 
46870 
46896 
46921 
46947 

88417 
88404 
88390 
88377 
88363 

88349 
88336 
88322 
88308 
88295 

48252 
48277 
48303 
48328 
48354 

48379 
48405 
48430 
48456 
48481 

87589 

87575 
87561 
87546] 
87532j 
87518) 
87504) 
87490 
87476 
87462 

49773 
49798 
49824 

49849 
49874 

49899 
49924 
49950 

49975 
50000 

86733 
86719 
86704 
86690 
86675 
86661 
86646 
86632 
86617 
86603 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Co&ine. 

Sine. 

Cosine. 

Sine. 

64° 

63° 

62° 

61° 

60° 

NATURAL  SINES  AND  COSINES. 

l 
2 
3 

4 
5 
6 
7 
8 
9 
10 

30° 

31° 

32° 

33° 

34° 

' 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

50000 
50025 
50050 
50076 
50101 
50126 
50151 
50176 
50201 
50227 
50252 

86603 
86588 

86573 
86559 

86544 
86530 

865*5 
86501 
86486 
86471 
86457 

5*5*9 
5*554 
5*579 
51604 
51628 

5*653 
5*678 

5*7*8 
5*753 

85717 
85702 
85687 
85672 

85657 
85642 

85627 
85612 

85597 
85582 

85567 

5*99* 

53017 

53066 
53°9* 
53**5 
53*40 
53*64 
53189 
53**4 
53*38 

84805 
84789 
84774 
84759 
84743 
84728 

84712 
84697 
84681 
84666 
84650 

54464 
54488 

545*3 

54537 

54586 
54610 
54635 
54659 
54683 
54708 

83867 
83851 

83835 
83819! 

83804 
83788 
83772 

83756 
83740 

83724 
83708 

559*9 

55943 
55968 

5599* 
56016 
56040 
56064 
56088 

561  12 
56136 
56160 

82904 

82887 
82871 
82855 
82839 
82822 
82806 
82790 
82773 
82757 
82741 

60 
59 
58 
57  ' 
56 
55 
54 
53 
52 
51 
50 

11 
12 

13 
14 
15 
16 
17 
18 
L9 
20 

50277 
50302 

50327 
5035.2 

50377 
50403 
50428 

5°453 
50478 

50503 

86442 
86427 
86413 
86398 
86384 
86369 

86354 
86340 
86325 
86310 

5*778 
5*803 
51828 
51852 
5*877 
51902 
5*9*7 

5*977 
52002 

8555* 
85536 
85521 
85506 
85491 

85476 
85461 
85446 
85431 
85416 

53*63 
53*88 
533** 
53337 

53386 

534** 
53435 
5346o 

53484 

84635 
84619 
84604 
84588 
84573 

84557 
84542 
84526 
84511 
84495 

5473* 
54756 
5478i 
54805 
54829 

54854 
54878 
54902 

549*7 

5495* 

83692 
83676] 
83660 

83645 
83629 

83613 
83597 
83581 

83565 
83549 

56184 
56208 
56232 
56256 
56280 

56305 
56329 

56353 

56377 
56401 

82724 
82708 
82692 
82675 
82659 
82643 
82626 
82610 
82593 
82577 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

38 
37 
36 
35 
34 
33 
32 
31 
30 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

~31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

11 

43 
44 
45 
46 
47 
48 
49 
50 

52 
53 
54 
55 

56 
57 
58 
59 
60 

50528 

50553 
50578 
50603 
5.0628 
50654 
50679 
50704 
50729 
50754 

86295 
86281 
86266 
86251 
86237 
86222 
86207 
86192 
86178 
.86163 

52026 
52051 
52076 
52101 
52126 

5**5* 

52175 
52200 
52225 
52250 

85401 

85385 
85370 
85355 
85340 
853*5 
853*0 
85*94 
85*79 
85264 

53509 
53534 
53558 

53583 
53607 

5363* 
53656 
53681 

53705 
5373° 

84480 
84464 
84448 
84433 
844*7 
84402 
84386 
84370 
84355 

54975 
154999 

55o*4 

55072 

55097 
55121 

55*45 
55*69 
55*94 
55218 
55242 
55266 
55291 
553*5 
55339 
55363 
55388 

55436 

83533 

83517 

83469 
83453 

83437 
83421 

83405 
83389 

564*5 
56449 

56473 
56497 
56521 

56545 
[56569 

!56593 
56617 
56641 

82561 

82544 
82528 
82511 
82495 
82478 
82462 
82446 
82429 
82413 

50779 
50804 
50829 
50854 
50879 
50904 
50929 

50954 
50979 
51004 

86148 
86133 
86119 
86104 
86089 
86074 
86059 
86045 
86030 
86015 

5**75 
52299 

5*3*4 
5*349 
5*374 

5*399 
5*4*3 
5*448 

5*473 
52498 

85249 
85234 
85218 
85203 
85188 

85173 
85*57 
85142 
85127 
85112 

53754 
53779 
53804 
53828 

53853 

53877 
53902 

539*6 
5395* 
53975 

843*4 
84308 
84292 

84*77 
84261 

84245 
84230 
84214 
84198 
84182 

83373 
83356 
83340 
833*4 
83308 

83292 

83260 

83*44 
83228 

156665  82396 

56689  82380 
56713  82363 
[56736  82347 
56760  82330 
5678482314 
56808  82297 
56832182281 
56856  82264 
56880  82248 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

51029 
5*054 
5*079 
51104 

5*i*9 

51179 

51204 
51229 
51254 

86000 

35985 
85970 

85956 
85941 
85926 
85911 
85896 
85881 
85866 

52522 
5*547 
5*57* 
5*597 
52621 

52646 

52696 
5*7*o 
5*745 

85096 
85081 
85066 
85051 
85035 
85020 
85005 
84989 

84974 
84959 

54000 
54024 

54°49 
54073 

54°97 

54*22 

54*46 

54*7* 
54*95 
5422Q 

54244 
54269 
54*93 
543*7 
5434* 
54366 
5439* 
544*5 
54440 

54464 

84167 
84151 

84*35 
84120 
84x04 
84088 

84057 
84041 
840*5 
84009 

83994 
83978 
83962 
83946 

83930 

839*5 
83899 
83883 
83867 

55484 
55509 
55'533 

55557 

5563° 
55654 
55678 

83212 
83*95 
83*79 
83163 

83*47 
83131 
83115 
83098 
83082 
83066 

56904 
56928 

56976 
57000 

57024 
i57°47 
57071 
57095 

57**9 

57*43 
157*67 
57191 
57215 
57*38 
57262 
57286 
573*0 
57334 
57358 

82231 
82214 
82198 
82181 
82165 
82148 
82132 
82115 
82098 
82082 

19 
18 
17 
16 
15 
14 
13 
12 
11  I 
10 

51279  85851 
5130485836 
5*3*9  85821 
51354  85806 
5*379  85792 
51404  85777 
5*4*9  85762 
51454185747 
51479  85732 
51504  85717 

5*770 

5*794 
52819 
52844 
52869 

5*893 
5*9*8 
52943 
5*967 
5*99* 

84943 
84928 

84913 
84897 
84882 

84866 
84851 
84836 
84820 
84805 

[55702 
55726 
[55750 
55775 
55799 

55847 
5587* 
55895 
1559*9 

83050 
83034 
83017 
83001 
82985 
82969 

82953 
82936 
82920 
82904 

82065 
82048 
82032 
82015 
8*999 
81982 
81965 
81949 
81932 
8*9*5 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

' 

59° 

58° 

57° 

|   56° 

55° 

94 


NATURAL  SINES  AND  COSINES. 

~b~ 

i 

2 
3 
4 
5 
6 
7 
8 
9 
10 

35° 

36° 

37° 

38° 

39° 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

6oT82 
60205 
60228 
60251 
60274 
60298 
60321 

60344 
60367 
60390 
60414 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

57358 
57381 

57405 
57429 

57453J 
57477 

57501 
57524 
57548 
57572 
57596 

8I9I5 
81899 
81882 
81865 
81848 
81832 
81815 
81798 
81782 
81765 
81748 

58779 
58802 
58826 

58849 
58873 
58896 
58920 
58943 
58967 
58990 
59014 

80902  i 
80885! 
80867 
80850! 
80833 
80816 

80799 
80782 
80765 
80748 
80730 

79864 
79846 

79829 
79811 

79793 
79776 

79758 
79741 
79723 
79706 
79688 

61566 
61589 
11612 

61635 
61658 
61681 

61704 

61726 

61749 
61772 
61795 

78801 
78783 
78765 

78747! 
787291 
78711 
78694 
78676 
78658 
78640 
78622 

&, 

62977 
63000 
63022 
63045 
63068 
63090 
63113 

63^58 

77696 
77678 
7766o 

77641 
77623 

77605 
77586 
77568 
7755° 
77531 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

57619 
57643 
57667 
57691 

57715 

57738 
57762 
57786 
57810 
57833 

81731 
81714 
81698 
81681 
81664 
81647 
81631 
81614 
81597 
81580 

59037 
59061 
59084 
59108 
59131 

59J54 
59178 
59201 

59225 
59248 

89713 
80696 
80679 
80662 
80644 
80627 
80610 
80593 
80576 
80558 

60437 
60460 
60483 
60506 
60529 

60553 
60576 

60599 
60622 
60645 

79671 
79653 
79635 
79618 
79600 

79583 
79565 
79547 
7953° 
79512 

61818 
61841 
61864 
61887 
61909 
61932 
61955 
61978 

62001 
62024 

78586 
78568 
78550 
78532 
78514 
78496 
78478 
78460 
78442 

63180 
63203 
63225 
63248 
63271 

63293 
63316 

63361 
6_3.3l3 
63406 
63428 
63451 
63473 
63496 

63518 

63585 
63608 

77513 
77494 
77476 

77458 
77439 

77421 
77402 

77384 
77366 

ZZ.347 

77329 

77310 
77292 
77273 
77255 
77236 
77218 
77199 
77181 

77l62 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

~3<r 

38 
37 
36 
35 
34 
33 
32 
31 
30 

21 

22 
23 
24 
25 
26 
27 
28 
29 
30 

57857 

579°4 
57928 
57952 
57976 
57999 
58023 

58047 
58070 

58094 
58118 
58141 
58165 
58189 
58212 
58236 
58260 
58283 

81563 
81546 
81530 
81513 
81496 

81479 
81462 
81445 
81428 
81412 

59272 
59295 
593l8 
59342 
59365 

59389 
59412 
59436 
59459 
159482 

80541 
80524 
80507 
80489 
80472 

80455 
80438 
80420 
80403 
80386 

60668 
60691 
60714 
60738 
60761 
60784 
60807 
60830 
60853 
60876 

79494 
79477 
79459 
79441 
79424 

79406 
79388 
79371 
79353 
79335 

62046 
62069 
62092 
62115 
62132 

62160 
62183 

62206 
62229 
62251 

78424 
78405 
78387 
78369 

78351 
78333 
78315' 
78297 
78279 
78261 

31 
32 
33 
34 
35 

36 
37 
38 
39 
40 

41 

42 
43 
44 
45 
46 
47 
48 
49 
50 

8l395i 
81378 
81361 
81344 
81327 
81310 
81293 
81276 
81259 
81242 

59506 
59529 
59552 
59576 
59599 
59622 
59646 
59669 

59693 
59716 

80368 
80351 
80334 
80316 
80299 
80282 
80264 
80247 
80230 
80212 

60899 
60922 
60945 
60968 
60991 
;  61015 
61038 
61061 
61084 
61  107 

79318 
79300 
79282 
79264 
79247 
79229 
79211 
79193 
79176 
79158 

62274 
62297 

62320 

62365 
62388 

62411 

62433 
62456 

62479 

782431 
78225; 
78206 
781881 
78170 
78152' 

78098 
78079 

63630 

63653 
63675 
63698 
63720 
63742 
63765 
63787 
63810 
63832 

77H4 
77125 
77107 

77088 

77070 

77051 
77033 
77014 

76996 
76977 

76959 
7-6940 
76921 
76903 
76884 

76866 
76847 
[76828 
76810 
76791 

29  I 
28 
27 
26 
25 
24 
23 
22 
21 
20 

5833° 
58354 
58378 
58401 

58425 

58449 
58472 

58496 
58543 

81225 
81208 
81191 
81174 
81157 
81140 
81123 
81106 
81089 
81072 

59739 

59786 

life 

59856 

59879 
59902 

59926 
59949 

80195 

80178 
80160 
i  80143 
,80125 
80108 
80091 
80073 
80056 
80038 

61130 

61176 
61199 
61222 
61245 
|6i268 
^61291 
61314 
6i337 

79140 
79122 
79105 
79087 
79069 
79051 

79033 
79016 

78998 
78980 

62502 

62524 
62547 
62570 
62592 
62615 
62638 

62660 

62683 

62706 

78061 
78043 
78o25 
78007 
77988 

77952 
77934 
77916 

77897 

63854 
63877 

63899 
63922 
63944 
63966 

63989 
64011 
64033 
64056 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

~F 
8 
7 
6 
5 
4 
3 
2 
1 
0 

51 

1  52 

54 
55 

56 

57 
58 
59 
60 

J58567 
158590 

58637 
58661 

58684 
58708 

(58755 
58779 

81055 
81038 
81021 
81004 
80987 
80970 
80953 
80936 
180919 
80902 

59972 

59995 
60019 
60042 
60065 
60089 
60112 
60135 
60158 
60182 

80021 
80003 
79986 
79968 
79951 

79934 
79916 

79899 

79881 

79864 

61360 
161383 
!  61406 
61429 
61451 
61474 

6,1497 
6*1520 

•^543 

!  61566 

78962 
78944 
78926 
78908 
78891 
78873 
78855 
78837 
(78819 
78801 

62728 
62751 

62774 

62796 
62819 
62842 
62864 
62887 
62909 

1  62932 
Cosine 

77879 
77861 

77843 
77824 
778o6 

77788 
77769 
77751 
77733 

64078  76772 
64100176754 
64123176735 
64145  76717 
64167  76698 

64190  76679 
64212  76661 
64234  76642 
64256  76623 
64279  76604 

/ 

Cosine 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Sine. 

Cosine. 

Sine. 

' 

54° 

63° 

52° 

61° 

60° 

95 


TXA 

TUB 

AZ,  J 

3IWI 

3S  A 

WD  < 

cosx: 

Eras. 

/ 

4 

0° 

4! 

1° 

4! 

2° 

4J 

j° 

4' 

I0 

/ 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

64279 
64301 
64323 
64346 
64368 
64390 
64412 

64435 
64457 
64479 
64501 

76604 
76586 

76567 
76548 
76530 
76511 

76492 
76473 
76455 
76436 
76417 

65606 
65628 
65650 

6567^ 
65694 
65716 

65738 
65759 
65781 
65803 
65825 

75471 
75452 
75433 
754H 
75395 
75375 
75356 
75337 
753l8 
75299 
75280 

66913 

*£93I 
66956 

66978 
66999 
67021 

67043 
07064 
67086 
67107 
67129 

743*4 
74295 
74276 
74256 

74237 
74217 
74198 
74178 
74159 
74139 
74120 

68200 
68221 
68242 
68264 
68285 
68306 
68327 

68349 
68370 
68391 
68412 

73135 
73116 
73096 
73076 
73056 
73036 
73016 
72996 
72976 
72957 
72937 

69466 
69487 
69508 
69529 
69549 
69570 
69591 
69612 
69633 
69654 
69675 

7*934 

71914 
71894 

71873 
71853 
71833 
71813 
71792 
71772 
71752 
71732 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

il 

$£ 

64635 
64657 

64679 
64701 
64723 

76398 
76380 
76361 
76342 
76323 

$u 

76267 
76248 
76229 

65847 
65869 
65891 
65913 
65935 
65956 
65978 
66000 
66022 
66044 

75261 
75241 
75222 
75203 
75184 
75165 
75146 
75126 
75107 
75088 

67151 
67172 
67194 
67215 
67237 
67258 
67280 
67301 
67323 
67344 

74100 
74080 
74061 
74041 
74022 
74002 

73983 
73963 

73944 
73924 

68434 

68455 
68476 

68497 
68518 

68539 
68561 
68582 
68603 
68624 

72917 
72897 
72877 
72857 
72837 

72817 
72797 
72777 
72757 
72737 

69696 
69717 
69737 
69758 
69779 

69800 
69821 
69842 
69862 
69883 

71711 
71691 
71671 
71650 
71630 

71610 

71590 
71569 

71549 
71529 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

6$t 

64790 
64812 
64834 
64856 
64878 
64901 
64923 
64945 

76210 

76192 
76173 

76154 
76135 

76116 

76097 
76078 
76059 
76041 

66066 
66088 
66109 
66131 
66153 

~175 
66197 

66218 
66240 
66262 

75069 

75°5° 
75030 
75011 
74992 

74973 
74953 
74934 
749  i  5 
74896 

67366 
67387 
67409 
67430 
67452 

67473 
67495 
67516 

67538 
67559 

73904 
73885 
73865 
73846 
73826 
73806 
73787 
73767 

73747 
73728 

68645 
68666 
68688 
68709 
68730 

68751 
68772 
68793 
68814 
68835 

72717 
72697 
72677 
72657 
72637 
72617 
72597 
72577 
72557 
72537 

69904 
69925 

*99i2 

69966 

69987 
70008 
70029 
70049 
70070 
70091 

71508 
71488 
71468 

7J447 
71427 

71407 
71386 
71366 
71345 
71325 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

64967 
64989 
65011 
65033 
65055 
65077 
65100 
65122 

!?5Iii 
65166 

76022 

76003 

75984 
75965 
75946 
75927 
75908 
75889 
75870 
75851 

66284 
66306 
66327 
66349 
66371 

66393 
66414 
66436 
66458 
66480 

74876 
74857 
74838 
74818 

74799 
7478o 
74760 

74741 
74722 

74703 

67580 
67602 
67623 
67645 
67666 
67688 
67709 
67730 
67752 
67773 

73708 
73688 
73669 

73649 
73629 

73610 
7359° 
7357° 
73551 
73531 

68857 
68878 
68899 
68920 
68941 
68962 
68983 
69004 
69025 
69046 

72517 
72497 
72477 

72457 
72437 

72417 

72397 
72377 

72357 
72337 

70112 

70132 

70153 
70174 
70195 
70215 
70236 

70257 
70277 
70298 

71305 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
71121 

29 
28 
27 
26 
25 
24 
23 
22 
21 
•20 

41 

42 
43 
44 
45 
46 
47 
48 
49 
50 

65188 
65210 
65232 

A5*5! 
65276 

65298 
65320 

6534? 
65364 
65386 

75832 
75813 
75794 
75775 
75756 

75738 

75719 
75700 
75680 
7566! 

66501 
66523 
66545 
66566 
66588 
66610 
66632 
66653 
66675 
66697 

74683 
74664 
74644 

74^5 
74606 

74586 
74567 
74548 
74528 

74509 

67795 
67816 
67837 
67859 
67880 
67901 
67923 
67944 
67965 
67987 

735" 
73491 
73472 
73452 
73432 

73413 
73393 
73373 
73353 
73333 

69067 
69088 
69109 
69130 
69151 
69172 
69193 
69214 
69235 
69256 

72317 
72297 
72277 
72257 
72236 
72216 
72196 
72176 
72156 
72136 

70319 

70339 
70360 
70381 
70401 
70422 
70443 
70463 
70484 
70505 

71100 
71080 
71059 
71039 
71019 
70998 
70978 
70957 
70937 
70916 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

65408 
65430 
65452 

65496 
65518 
65540 
65562 

65584 
65606 

75642 
75623 
75604 

75585 
75566 

75547 
75528 

755°9 
7549° 
75471 

66718 
66740 
66762 
66783 
66805 
66827 
66848 
66870 
66891 
66913 

74489 
74470 

74451 
7443  * 
74412 

74392 
74373 
74353 
74334 
743H 

68008 
68029 
68051 
68072 
68093 
68115 
68136 

68157 
68179 
68200 

733H 
73294 

73274 
73254 
73234 
73215 
73195 
73J75 
73155 
73135 

69277 
69298 
69319 
69340 
69361 
69382 

69403 
69424 

69445 
69466 

72116 
72095 
72075 
72055 
72035 
72015 
71995 
71974 
71954 
71934 

70525 
70546 
70567 
70587 
70608 
70628 
70649 
70670 
70690 
70711 

70896 
70875 
70855 
70834 
70813 

70793 
70772 

70752 
70731 
70711 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

/ 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

Cosine. 

Sine. 

4 

9° 

4* 

*° 

4' 

1° 

4( 

J° 

4£ 

»° 

96 


TABLE   OF   CHORDS. 


A  TABLE    OF  CHORDS. 


M. 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

M. 

~T 

.0000 

•oi75 

•0349 

.0524 

.0698 

.0872 

.1047 

.1221 

•1395 

o 

5 

.0015 

.0189 

.0364 

.0538 

.0713 

.0887 

.1061 

.1235 

.1410 

5 

10 

.0029 

.0204 

.0378 

•°553 

.0727 

.0901 

.1076 

.1250 

.1424 

10 

15 

.0044 

.0218 

.0393 

.0567 

.0742 

.0916 

.1090 

.1265 

.1439 

15 

20 

.0058 

.0233 

.0407 

.0582 

.0756 

.0931 

.1105 

.1279 

•1453 

20 

25 

.0073 

.0247 

.0422 

.0596 

.0771 

.0945 

.1119 

.1294 

.1468 

25 

30 

.0087 

.0262 

.0436 

.0611 

.0785 

.0960 

.1134 

.1308 

.1482 

30 

35 

.0102 

.0276 

.0451 

.0625 

.0800 

.0974 

.1148 

.1323 

.1497 

35 

40 

.OIl6 

.0291 

.0465 

.0640 

.0814 

.0989 

.1163 

•1337 

.1511 

40 

45 

.0131 

.0305 

.0480 

.0654 

.0829 

.1003 

.1177 

•1352 

.1526 

45 

50 

.0145 

.0320 

.0494 

.0669 

.0843 

.1018 

.1192 

.1366 

.1540 

50 

55 

.Ol6o 

.0335 

.0509 

.0683 

.0858 

.1032 

.1206 

.1381 

•1555 

55 

60 

•°I75 

•°349 

.0524 

.0698 

.0872 

.1047 

.1221 

•1395 

.1569 

60 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

0 

.1569 

.1743 

.1917 

.2091 

.2264 

•2437 

.2611 

.2787 

.2956 

~0~ 

5 

.1584 

.1758 

.1931 

.2105 

.2279 

.2452 

.2625 

.2798 

.2971 

5 

10 

.1598 

.1772 

.1946 

.2119 

.2293 

.2466 

.2639 

.2812 

.2985 

10 

15 

.1613 

.1787 

.1960 

.2134 

.2307 

.2481 

.2654 

.2827 

.2999 

15 

20 

.1627 

.1801 

•!975 

.2148 

.2322 

.2495 

.2668 

.2841 

.3014 

20 

25 

.1642 

.1816 

.1989 

.2163 

.2336 

.2510 

.2683 

.2855 

.3028 

25 

30 

.1656 

.1830 

.2004 

.2177 

.2351 

.2524 

.2697 

.2870 

.3042 

30 

35 

.1671 

.1845 

.2018 

.2192 

.2365 

.2538 

.2711 

.2884 

.3057 

35 

40 

.1685 

.1859 

.2033 

.2206 

.2380 

•2553 

.2726 

.2899 

.3071 

40 

45 

.j-oo 

.1873 

.2047 

.2221 

.2394 

.2567 

.2740 

.2913 

.3086 

45 

50 

.1714 

.1888 

.2062 

.2235 

.2409 

.2582 

•^755 

.2927 

.3100 

50 

55 

.^729 

.1902 

.2076 

.2250 

.2423 

.2596 

.2769 

.2942 

•3"4 

55 

60 

•1743 

.1917 

.2091 

.2264 

.2437 

.2611 

.2783 

.2956 

.3129 

60 

98 


TABLE  OF  CHORDS. 

M.I 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

M. 

0 

.3129 

.3301 

•3473 

•3645 

.3816 

•3987 

.4158 

•43*9 

•4499 

0 

5 

•3H3 

.3487 

.3659 

.3830 

.4002 

.4172 

•4343 

•45  1  3 

5 

10 

•3*57 

.3330 

.3502 

.3673 

•3845 

.4016 

.4187 

•4357 

•45*7 

10 

15 

.3172 

•3344 

.3516 

.3688 

•3859 

.4030 

.4201 

•4371 

•454* 

15 

20 

.3186 

•335* 

•353° 

.3702 

•3873 

.4044 

.4215 

.4386 

.4556 

20 

25 

.3200 

•3373 

•3545 

.3716 

.3888 

.4059 

.4229 

.4400 

.4570 

25 

30 

•3*'5 

•3387 

•3559 

•373° 

.3902 

•4073 

•4*44 

•44  *  4 

•4584 

30 

35 

.3229 

.3401 

•3573 

•3745 

.3916 

.4087 

.4258 

.4428 

•4598 

35 

40 

•3*44 

.3416 

•3587 

•3759 

•393° 

.4101 

.4272 

•444* 

.4612 

40 

45 

.3258 

•343° 

.3602 

•3773 

•3945 

.4116 

.4286 

•445  6 

.4626 

45 

50 

•3*7* 

•3444 

.3616 

•3788 

•3959 

.4130 

.4300 

.4471 

.4641 

50 

55 

•3*87 

•3459 

•363° 

.3802 

•3973 

.4144 

•43  1  5 

•4485 

.4655 

55 

60 

•33°i 

•3473 

.3645 

.3816 

•3987 

.4158 

•43*9 

•4499 

.4669 

60 

27° 

28° 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

0 

.4669 

.4838 

.5008 

•S1?^ 

•5345 

.5513 

.5680 

•5847 

.6014 

0 

5 

10 

.4683 
.4697 

•4853 
.4867 

.5022 
.5036 

.5190 
.5204 

•5359 
•5373 

•55*7 
•5541 

.5694 

•5708 

.5861 

•5875 

.6028 
.6042 

5 

10 

15 

.4711 

.4881 

.5050 

.5219 

•5387 

•5555 

•57** 

..5889 

.6056 

15 

20 

•47*5 

•4895 

.5064 

•5*33 

.5401 

•5569 

.5736 

•59°3 

.6070 

20 

25 

.4740 

.4909 

.5078 

•5*47 

•5415 

•5583 

•575° 

•5917 

.6083 

25 

30 

•4754 

•49*3 

.5092 

.51*1 

•54*9 

•5597 

.5764 

•5931 

.6097 

30 

35 

.4768 

•4937 

.5106 

•5*75 

•5443 

.5611 

•5778 

•5945 

.6m 

35 

40 

.4782 

•495  i 

.5120 

.5289 

•5457 

.S62S 

•579* 

•5959 

.6125 

40 

45 

.4796 

.4965 

•S'34 

.5303 

.5471 

•5638 

.5806 

•597* 

.6139 

45 

50 

.4810 

•4979 

.5148 

•5317 

•5485 

.5652 

.5820 

.5986 

•6153 

50 

55 
60 

.4824 
.4838 

•4994 
.5008 

.5162 

•5331 
•5345 

•5499 
•5513 

.5666 
.5680 

•5833 
•5847 

.6000 
.6014 

.6167 
.6180 

55 
60 

36° 

37° 

38° 

39° 

40° 

41° 

42° 

43° 

44° 

0 

.6180 

.6346 

.6511 

.6676 

.6840 

.7004 

.7167 

•733° 

•749* 

0 

5 

.6194 

.6360 

.6525 

.6690 

.6854 

.7018 

.7181 

•7344 

.7506 

5 

10 

.6208 

•6374 

.6539 

.6704 

.6868 

.7031 

•7i95 

•7357 

10 

15 

.6222 

.6387 

•6553 

.6717 

.6881 

.7045 

.7208 

•7371 

•7533 

15 

20 

.6236 

.6401 

.6566 

.6731 

.6895 

.7059 

.7222 

•7384 

.7546 

20 

25 

.6249 

.6415 

.6580 

.6745 

.6909 

.7072 

•7*35 

•7398 

.7560 

25 

30 

.6263 

.6429 

.6594 

•6758 

.6922 

.7086 

•7*49 

.7411 

•7573 

30 

35 

.6277 

.6443 

.6608 

.6772 

.6936 

.7099 

.7262 

•74*5 

•7586 

35 

40 
45 

.6291 
.6305 

.6456 

.6470 

.6621 

.6786 
.6799 

.6950 
.6963 

•7113 
.7127 

.7276 
.7289 

•743s 
•745* 

.7600 
.7613 

40 
45 

50 

.6319 

.6484 

.6649 

.6813 

.6977 

.7140 

•73°3 

.7465 

.7627 

50 

55 
60 

.6332 
.6346 

.6498 
.6511 

.6662 
.6676 

.6827 
.6840 

.6991 

.7004 

•7i54 
.7167 

.7316 
•733° 

•7479 
•749* 

.7640 
.7654 

55 
60 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

52° 

53° 

0 
5 

•7654 
.7667 

•7815 
.7828 

•7975 
.7988 

•813? 
.8148 

.8294 
.8307 

.8452 
.8466 

.8610 
.8623 

.8767 
.8780 

.8924 
•8937 

0 
5 

10 

.7681 

.7841 

.8002 

.8161 

.8320 

•8479 

.8636 

•8794 

.8950 

10 

!  15 

.7694 

•7855 

.8015 

.8l7C 

•8334 

.8492 

.8650 

.8807 

.8963 

15 

20 

.7707 

.7868 

.8028 

.8188 

•8347 

.8coc 

.8663 

.8820 

.8976 

20 

25 

•77*i 

.7882 

.8042 

.8201 

.8360 

•8518 

.8676 

•8833 

.8989 

25 

30 
35 

•7734 
.7748 

•7895 
.7908 

!8oto 

.8214 

.8228 

18386 

•8531 

•8545 

.8689 

.8702 

.8846 
.8859 

.9002 
.9015 

30 
35 

40 

.7761 

.7922 

.8082 

.8241 

.8400 

.8715 

.8872 

.9028 

40 

45 

•7774 

•7935 

.8o9<; 

.8254 

.8413 

.8571 

.8728 

.8885 

.9041 

45 

50 

55 

.7788 
.^801 

•7948 
.7962 

.8108 
.8121 

.8267 
.8281 

.8426 
•8439 

.8584 
•8597 

.8741 
•8754 

.8898 
.8911 

.9054 
.9067 

50 
55 

60 

•7815 

•7975 

.8135 

.8294 

.8452 

.8610 

.8767 

.8924 

.9080 

60 

99 


TA 

BLE 

or  < 

;HOR; 

DS. 

M. 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

61° 

62° 

M. 

o" 

.9080 

•9*35 

•9389 

•9543 

.9696 

.9848 

.0000 

.0151 

.0301 

T 

5 

•9°93 

.9248 

.9402 

•9556 

.9709 

.9861 

.0013 

.0163 

•°3I3 

5 

10 

.9106 

.9261 

.9415 

.9569 

.9722 

.9874 

.0025 

.0176 

.0326 

10 

15 

.9119 

.9274 

.9428 

.9581 

•9734 

.9886 

.0038 

.0188 

•0338 

15 

20 
25 

.9132 
.9145 

.9287 
.9299 

.9441 
•9454 

•9594 

.9607 

•9747 
.9760 

•9899 
•9912 

.0050 
.0063 

.0201 
.0213 

•O35i 
•0363 

20 
25 

30 

•9i57 

.9312 

•9466 

.9620 

.9772 

.9914 

.0075 

.0226 

•°375 

30 

35 

.9170 

•93*5 

•9479 

•9633 

.9785 

•9937 

.0088 

.0238 

35 

40 

.9183 

•9338 

.9492 

•9645 

.9798 

•995° 

.0101 

.0251 

.0400 

40 

45 

.9196 

•9351 

.9505 

.9658 

.9810 

.9962 

.01  13 

.0263 

.0413 

45 

50 

.9209 

.9364 

.9518 

.9671 

.9823 

•9975 

.0126 

.0276 

.0425 

50 

55 

.9222 

•9377 

•953° 

.9683 

.9836 

.9987 

.0138 

.0288 

.0438 

55 

60 

•9*35 

•9389 

•9543 

.9696 

.9848 

1.  0000 

.0151 

.0301 

.0450 

60 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

70° 

71° 

~T 

1.0450 

.0598 

1.0746 

.0893 

.1039 

.1184 

.1328 

.1472 

.1614 

0 

5 

1.0462 

.0611 

1.0758 

.0905 

.1051 

.1196 

.1340 

.1483 

.1626 

5 

10 

1.0475 

.0623 

1.0771 

.0917 

.1063 

.1208 

•!352 

•'495 

.1638 

10 

15 

1.0487 

.0635 

1.0783 

.0929 

•1075 

.1220 

.1364. 

^S0? 

.1650 

15 

20 

1.0500 

.0648 

1.0795 

.0942 

.1087 

.1232 

.1376 

.1519 

.1661 

20 

25 

1.0512 

.0660 

1.0807 

.0954 

.1099 

.1244 

.1388 

•i53i 

.1673 

25 

30 

1.0524 

.0672 

1.0820 

.0966 

.mi 

.1256 

.1400 

•'543 

.1685 

30 

35 
40 

1-0537 
1.0549 

.0685 
.0697 

1.0832 
1.0844 

.0978 
.0990 

.1123 
.1136 

.1268 
.1280 

.1412 

.1424 

•1555 
.1567 

.1697 
.1709 

35 
40 

45 

1.0561 

.0709 

1.0856 

.1002 

.1148 

.1292 

.1436 

•1579 

.1720 

45 

50 

1.0574 

.0721 

1.0868 

.1014 

.1160 

.1304 

.1448 

.1590 

.1732 

50 

55 

1.0586 

.0734 

1.0881 

.1027 

.1172 

.1316 

.1460 

.1602 

.1744 

55 

60 

1.0598 

.0746 

1.0893 

.1039 

.1184 

.1328 

.1472 

.1614 

.1756 

60 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

80° 

T 

.1756 

.1896 

.2036 

.2175 

.2313 

1.2450 

1.2586 

.2722 

.2856 

0 

5 

.1767 

.1908 

.2048 

.2187 

.2325 

1.2462 

1.2598 

•2733 

.2867 

5 

10 

.1779 

.1920 

.2060 

.2198 

.2336 

.2473 

1.2609 

.2744 

.2878 

10 

15 

.1791 

.1931 

.2071 

.2210 

.2348 

.2484 

1.2620 

•2755 

.2889 

15 

20 

.1803 

.1943 

.2083 

.2221 

•2359 

.2496 

1.2632 

.2766 

.2900 

20 

25 

.1814 

•'955 

.2094 

.2233 

.2370 

.2507 

1.2643 

.2778 

.291  1 

25 

30 

.1826 

.1966 

.2106 

.2244 

.2382 

.2518 

1.2654 

.2789 

.2922 

30 

35 

.1838 

.1978 

.2117 

.2256 

.2393 

.2530 

1.2665 

.2800 

•2934 

35 

40 

.1850 

.1990 

.2129 

.2267 

.2405 

.2541 

1.2677 

.2811 

.2945 

40 

45 

.1861 

.2001 

.2141 

.2279 

.2416 

.2552 

1.2688 

.2822 

.2956 

45 

50 

•1873 

.2013 

.2152 

.2290 

.2428 

.2564 

1.2699 

.2833 

.2967 

50 

55 

.1885 

.2025 

1.2164 

.2302 

•2439 

•2575 

1.2710 

.2845 

.2978 

55 

60 

.1896 

.2036 

1.2175 

.2313 

.2450 

.2586 

1.2722 

.2856 

.2989 

60 

81° 

82° 

83° 

84° 

85° 

86° 

87° 

88° 

89° 

~T 

.2989 

.3121 

.3252 

1-3383 

•3512 

.3640 

•3767 

•3893 

.4018 

~F 

5 

.3000 

•313* 

.3263 

1-3393 

•3523 

•3651 

.3778 

•39°4 

.4029 

5 

10 

.3011 

•3*43 

•3274 

•34°4 

•3533 

1  .3661 

.3788 

•39H 

•4°39 

10 

15 

.3022 

•3*54 

.3285 

•34*5 

•3544 

.3672 

•3799 

•39^5 

.4049 

15 

20 

•3°33 

•3i65 

.3296 

.3426 

•3555 

.3682 

.3809 

•3935 

.4060  ! 

20 

25 

•3°44 

.3176 

•33°7 

•3437 

•3565 

•3693 

.3820 

•3945 

.4070 

25 

30 

•3°55 

•3187 

•33i8 

•3447 

•3576 

•37°4 

•383° 

•3956 

.4080 

30 

35 

.3066 

.3198 

.3328 

•3458 

•3587 

•37H 

.3841 

.3966 

.4091 

35 

40 

.3077 

.3209 

•3339 

•3469 

•3597 

•37^5 

•3851 

•3977 

.4101  1 

40 

45 

.3088 

.3220 

•335° 

.3480 

.3608 

•3735 

.3862 

•3987 

.4111 

45 

50 

.3099 

•3231 

•3361 

•349° 

.3619 

•3746 

.3872 

•3997 

.41221 

50 

55 

.3110 

.3242 

.3372 

•35°i 

.3629 

•3757 

•3883 

.4008 

.4132! 

55 

60 

.3121 

•3*5* 

.3383 

.3512 

.3640 

.3767 

•3893 

.4018 

.4142 

60 

100 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


SEP  15  1940 

3       '        "    - 

1      r»**-*i 

OCT  241941 

w  ftl   IJ 
M|iiUJULJE 

JW»  2<*  m, 

*oiW531  >:i 

u             ,i& 
04951= 

DEC  26  1944 

*v 

LIBRAPV 

jftN    28  1946 

JUL  2  1  1955 

NOV  9     1946 

JUL211955LU 

icocr 

!  -i;^  ^i 

,  if.  -v  ot  k  .. 

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LD  21-100m-7,'39(402s) 

F3525. 


U.C.BERKELEYLIBRA1R1ES 


•  - 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


